Scroll 1
Aranfan
Team Plasma Grunt
It's time to learn Calculus bitches! The Infinitesimal kind, as Euler and Leibniz intended!
I shall assume that everyone reading this has a solid grasp on algebra and knows a bit of geometry.
In euclidean geometry you can draw a line between any two points, and extend said line indefinitely. You can draw a circle with the center at any point and the edge at any other.
So you pick two points and say the line between them is length 1. You extend this line indefinitely in both directions. Label one point (0,0) and the other (1,0) Use circles to mark off points at regular intervals of unit length. The point one unit length away from (0,0) on the opposite side from (1,0) is (-1,0). You now have directions. Call this line the X-Axis. A point on the X-axis can be uniquely specified by (x,0) where x is any real number.
By Euclid 1-11, it is possible to draw a line perpendicular to the X-Axis at (0,0). Do so to create the Y-Axis. Use circles to mark off points of unit length on it, label points on the Y-Axis (0, y) where y is any real number. Choose a (0,1) and (0,-1) for direction. It doesn't technically matter which side you pick, but convention is to say (0,1) is "above" the X axis.
By Euclid 1-12 it is possible to drop a perpendicular from any point not on a line to the line in question. By dropping a perpendicular from any point in the plane we can see where on the X and Y axes those perpendiculars will fall. and we can call that point (x₁,y₁) where x₁ and y₁ are the length from the origin to the perpendicular dropped from the point to the axis in question. It is obvious also that if you make perpendiculars at (x₁,0) and (0,y₁) they will intersect at that same point. Thus we are able to uniquely identify any point with an ordered pair of numbers (x,y).
By the Pythagorean Theorem the distance s between any two points is:
A little work will show that a line between any two points can be described by an equation:
And a circle with a center at a given point and a radius r is given by
With this we can use algebra to do geometry. We can find lengths and intersections and tangents of circles and all the other things Euclid could do in two dimensions. Adding a Z-axis lets us do solid geometry, once we figure out the equations for a plane or a sphere. Be we also now have the tools to go beyond Euclid. To describe curves impossible to create with only a compass and straight edge. The Conic Sections can be described with only equations of degree 2, and Equations of degree 3 can describe curves never conceived of by the great Hellenistic Geometers of old.
The obvious questions arise however... can we find the tangents and areas of these new figures?
To draw a secant line on a curve is dead simple, you take two points on the curve, and draw a line, and bam your done. Tangents are much, much harder. This is because with a Tangent you only have one point, because a tangent is a line that only touches a curve at one point. The equation for a line above collapses into 0=0 if both points are the same. You can't draw a line with only one point! Right?
Well. Not quite. The equation for a line I gave above is the most general form, but it's not the most common. The most common form of the equation of a line is
The trick here is that for a line, m will be the same no matter what two points you pick on the line. The ratio of rise (change in y coordinate) over run (change in x coordinate) will always be the same no matter where on the line you are. So if we had some way to find m, independently of a second point... we'd be able to draw a line anyway. But we can't just choose a value for m willy nilly if we want to make a tangent. So we are once again stuck. We can draw a line with only one point if we have a slope, but the constraints on a tangent means we need to find m and the only way we know of spits out division by zero if we only give it one point.
So what if we didn't?
Have you ever noticed that the more sides a regular polygon has the more it looks like a circle? That the bigger the circle is, the more a small enough arc of it looks like a line? What if we zoomed in really far?
It is time to introduce the star of the infinitesimal calculus, the differential: df.
Let f be a function of a variable q and every other variable is in some way related to q but we don't know how. Then
Where the "d" stands for difference, and isn't a new variable but rather attaches to an existing one. You can't cancel out "d", but you can cancel out "dx" or "dy" or "du" or etc.
So what makes dx so special? Right now it just seems like a normal difference you could find using two points. The difference is that h is small, really small, infinitely small in fact. Which means that for any case where you are adding or subtracting h from an expression that doesn't involve h at all... it's negligible.
Lets use our new friend the differential to calculate the tangents to a simple curve like the parabola y=x^2. We see that rise over run is just a quotient of differences, so we want to find dy/dx:
Now of course this isn't the slope of a tangent. This is a secant's... but note that every secant's slope where the points are only dx apart differ by dx. The appreciable part, if you will, of the slope of all these secants are... 2x. All of them. It doesn't matter which infinitesimal we use for h or dx, positive or negative. They all have the same appreciable part. So we can just round off the infinitesimal.
The equation for a tangent to a parabola y=x^2 at a point is
It has to be this, because it can't be anything else. A formal proof for every tangent calculation of this sort, that it can't be more nor less, would be exhausting, but in principle possible. I'm not going to bother doing it. However, note that the condition that the appreciable part not depend on the infinitesimal increment is very important. If the appreciable part of the ratio dy/dx in any way depends on dx, then we can't round off the infinitesimals and we can't infer a unique tangent line from the secants. This matters in cases where there is a sharp turn in the curve, like the absolute value at 0, or where a curve intersects itself.
Note that we can derive some rules for the differential. Applying d to a variable that is complicated enough can be broken down into easier steps.
By the binomial theorem, for integers you can see that:
To show this rule works for all exponents needs the use of the logarithm, but it does, in fact, work for all real numbers n.
The constant rule is that da is 0 when a is a constant. dπ? 0. This is clear because if f(x)=5, then it doesn't matter what the increment is, it'll be 5-5=0. Likewise d distributes over addition and subtraction. The more interesting rules to derive are the product and quotient rules
and:
Where the vdv is discarded as negligible compared to v^2 and dudv is likewise discarded as negligible compared to udv+vdu. The chain rule, on the other hand is just multiplication by 1:
Of course, once you are able to find tangents you can use them for all sorts of things! The most consistently useful of these is finding the highest and lowest values a of given curve. You just find where the differential quotient is zero and you have either a minimum or a maximum. The second derivative test can be used to find out which is which:
Which makes sense since the second derivative tells you if the curve is concave up or concave down by its sign. If the second derivative is also zero, such as with y=x^3 at 0, then things have gone strange and the point in question may be neither maxima nor minima.
As an aside, I'd like to complain about prevailing notation. The most common way to write the second derivative of y wrt x is:
But this is wrong. The first derivative is a quotient, so you need to use the quotient rule:
Which only reduces to the first notation if dx is a constant and ddx=0 by the constant rule. Unlike the first derivative, the typical notation of the second derivative will give you wrong answers if you treat it as an algebraically separable fraction, especially if the hidden assumption of x being the independent variable is violated. But the notation that uses the quotient rule does work out if treated as separable algebraic fractions.
Unlike finding tangents, areas are easy to find... well they're easy to approximate. To find the area between a curve and the x axis between two points you just need to draw a bunch of rectangles that are very thin, and add up all the rectangles. The thinner the rectangles the better the approximation. Of course, you can't just have zero width, then you would just have a bunch of lines. In order for the dimensions to work out, the rectangles need to have a width.
Anyway, this is how the area between a curve and the x axis, and the lines x=a and x=b, is denoted in the calculus
The fancy capital S stands for the infinite sum of all the rectangles of hight y and width dx between x=a and x=b. If you added up all those rectangles you would get the actual area... plus or minus an infinitesimal error term that could be rounded off, just like with the derivative. Note that this gives you the signed area. If the y value is negative, then the rectangle will have a negative sign attached to it. Likewise the dx has a direction, if you swap the bounds of integration you are flipping the sign of the dx and get a negative sign infront of every rectangle, which can then be pulled out of the integral by distributivity.
But this is all just academic theory. Adding up all those rectangles of y*dx would take literally forever.
Unless there was a trick. It doesn't actually make intuitive sense for there to be a trick. Areas and Tangents don't seem like they should have anything to do with each other. But they do. It turns out that:
Is, under certain conditions, a telescoping sum. i.e.
This works because the differential is a difference
And therefore the Infinite Sum and the Infinitesimal Difference are inverse operations, just like addition and subtraction are. Indeed, the integral is a sum of products and the derivative a quotient of differences. So long as the function is continuous, that is an infinitesimal dx corresponds to an infinitesimal dy, the integral of a derivative will telescope and collapse into an easy finite difference. Thus, so long as you don't try to integrate across a discontinuity, you can find areas exactly as long as you can find an anti-derivative. An example of where this condition fails is y=(1/x)^2 at 0. Infinitesimal dx corresponds to infinite dy, and you get plainly and obviously wrong answers if you try to treat the integral as a telescoping sum across zero.
So you're in business, so long as you can find an anti-derivative, you can find the area under a curve. The issue is that symbolic integration can sometimes be impossible using only algebraic functions. The anti-derivative of y=1/x cannot be expressed as a finite combination of additions, subtractions, multiplications, divisions, and root extractions. Even adding in the trig functions, their inverses, and the exponential and logarithmic functions only pushes the problem back. These functions are typically called the "elementary functions", and they are not closed under anti-differentiation. Still, it is useful to try and find symbolic anti-derivatives in terms of the elementary functions, because where we can find such expressions we can evaluate areas exactly.
Because mathematicians never let a symbol go to waste, the notation for anti-differentiation is exactly the same as for the integral, just without the bounds of integration.
Where the process of differentiating can be almost mechanical, the art of finding anti-derivatives is much fuzzier. Most methods of trying to find anti-derivatives is based on reversing the differentiation rules. Like the derivative, the integral distributes over addition and subtraction.
Integration by Parts is just the product rule in reverse:
Where figuring out which parts of the integrand to make u and which to make dv in order to make it easier rather than harder to find an anti-derivative is mostly a matter of trained intuition. There is a lot of depth to doing integration by parts correctly. You can even get taylor series out of integration by parts and the fundamental theorem of calculus. As an aside, if that second line reminds you of path integrals, by the way, there's a reason for that.
Integration by Substitution is literally just the chain rule in reverse. With Trig Substitution just being fancy with u-sub
Meanwhile, the power rule is again just the differential power rule in reverse:
Note that due to how any constant becomes zero when differentiated, when finding anti-derivatives you need to add a constant term to the expression. Which constant? We don't know enough to say unless given more information. So it is typically just noted as +C, and left as is.
There are a number of functions in math that "transcend" algebra. Most functions are transcendental, but the ones that actually are commonly encountered are fairly few. The Trig Functions, the Logarithmic Functions, and their inverses the arctrig functions and the exponential functions are the ones that actually were encountered before the invention of the Calculus dramatically expanded the range of functions people interacted with. So in this final part I'm going to show how to differentiate these functions from first principles... well the trig functions and the exponential and log anyway. The inverse trig functions are left as an exercise to the readers.
First is the trig functions. We only actually need one of them, if you know how to differentiate sine or cosine then the differentiation rules will let you differentiate any of the trig functions. But I'll go over both sine and cosine at the same time, because it's easy to do both at once when your working with differentials.
First we define the sine and cosine functions geometrically. Given a circle of radius r centered at the origin
Where (x,y) is the point on the circle where the ray from the origin at angle θ from the x axis intersects the circle. Letting s be the arclength, then from the definition of the circle and radians we know the following:
This is enough to do some algebra:
Looking at the circle we see that when a positive change in angle happens the sign of dy is the same as x, and the sign of dx is opposite y. Keeping in mind also that r is constant and can be moved in and out of the differential freely:
The logarithm was invented for a practical purpose: addition is easier than multiplication, so if we had a way to turn multiplication into addition we could spend less time in tedious computation and more time doing the interesting stuff. As such a logarithmic function is any non-zero function that obeys the following relationship:
This is where we bring back the idea that the Integral is an area and not just the reverse of differentiation. Back before the days of calculus, when a master was explaining his findings on the quadrature of the hyperbola y=1/x, his student noticed the following:
Which is to say that the function for the area under the curve of the hyperbola, measured starting from x=1, is given by a logarithmic function. Presently we don't know the base of the logarithm, but we do know that the derivative of this logarithm must be 1/x. Which is interesting, because a logarithmic function in any base is a constant multiple of any other base's logarithm. So the lack of any multiple means that this logarithm is in some sense natural, and it's corresponding exponential function must also be natural in the same sense. Lets call this "natural logarithm" ln(x).
The derivative of the inverse function is the reciprocal of the derivative. This makes sense because the derivative of y wrt x is the change in y divided by the change in x. Thus if you divide dx by dy instead it's like flipping the axes. Thus, for the exponential function of the same base, call that base "e" for now:
The differentials of the inverse trig functions can be found in the same way, which is left as an exercise to the reader.
This finally lets us prove the power rule for all real numbers, rational and irrational:
And to round us off, here's the functional power rule:
I shall assume that everyone reading this has a solid grasp on algebra and knows a bit of geometry.
PART 1: COORDINATE GEOMETRY
In euclidean geometry you can draw a line between any two points, and extend said line indefinitely. You can draw a circle with the center at any point and the edge at any other.
So you pick two points and say the line between them is length 1. You extend this line indefinitely in both directions. Label one point (0,0) and the other (1,0) Use circles to mark off points at regular intervals of unit length. The point one unit length away from (0,0) on the opposite side from (1,0) is (-1,0). You now have directions. Call this line the X-Axis. A point on the X-axis can be uniquely specified by (x,0) where x is any real number.
By Euclid 1-11, it is possible to draw a line perpendicular to the X-Axis at (0,0). Do so to create the Y-Axis. Use circles to mark off points of unit length on it, label points on the Y-Axis (0, y) where y is any real number. Choose a (0,1) and (0,-1) for direction. It doesn't technically matter which side you pick, but convention is to say (0,1) is "above" the X axis.
By Euclid 1-12 it is possible to drop a perpendicular from any point not on a line to the line in question. By dropping a perpendicular from any point in the plane we can see where on the X and Y axes those perpendiculars will fall. and we can call that point (x₁,y₁) where x₁ and y₁ are the length from the origin to the perpendicular dropped from the point to the axis in question. It is obvious also that if you make perpendiculars at (x₁,0) and (0,y₁) they will intersect at that same point. Thus we are able to uniquely identify any point with an ordered pair of numbers (x,y).
By the Pythagorean Theorem the distance s between any two points is:
LaTeX:
\[
s=+ \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}
\]
LaTeX:
\[
(y - y_1)(x_1 - x_2)=(y_1 - y_2)(x-x_1)
\]
LaTeX:
\[
(x-x_1)^2 + (y-y_1)^2 = r^2
\]
With this we can use algebra to do geometry. We can find lengths and intersections and tangents of circles and all the other things Euclid could do in two dimensions. Adding a Z-axis lets us do solid geometry, once we figure out the equations for a plane or a sphere. Be we also now have the tools to go beyond Euclid. To describe curves impossible to create with only a compass and straight edge. The Conic Sections can be described with only equations of degree 2, and Equations of degree 3 can describe curves never conceived of by the great Hellenistic Geometers of old.
The obvious questions arise however... can we find the tangents and areas of these new figures?
PART 2: TANGENTS
To draw a secant line on a curve is dead simple, you take two points on the curve, and draw a line, and bam your done. Tangents are much, much harder. This is because with a Tangent you only have one point, because a tangent is a line that only touches a curve at one point. The equation for a line above collapses into 0=0 if both points are the same. You can't draw a line with only one point! Right?
Well. Not quite. The equation for a line I gave above is the most general form, but it's not the most common. The most common form of the equation of a line is
LaTeX:
\[
(y-y_1)=m(x-x_1) \\ m=\frac{y_1 - y_2}{x_1 - x_2}
\]
So what if we didn't?
Have you ever noticed that the more sides a regular polygon has the more it looks like a circle? That the bigger the circle is, the more a small enough arc of it looks like a line? What if we zoomed in really far?
It is time to introduce the star of the infinitesimal calculus, the differential: df.
Let f be a function of a variable q and every other variable is in some way related to q but we don't know how. Then
LaTeX:
\[
df=f(q+h)-f(q) \\ f(q+h)=f(q)+df \\ df=(f(q)+df)-f(q)
\]
So what makes dx so special? Right now it just seems like a normal difference you could find using two points. The difference is that h is small, really small, infinitely small in fact. Which means that for any case where you are adding or subtracting h from an expression that doesn't involve h at all... it's negligible.
Lets use our new friend the differential to calculate the tangents to a simple curve like the parabola y=x^2. We see that rise over run is just a quotient of differences, so we want to find dy/dx:
LaTeX:
\[
y=x^2 \Rightarrow (y+dy)-y=(x+dx)^2-x^2 \\ dy=x^2+2xdx+(dx)^2-x^2 \\dy=2xdx+(dx)^2 \\ \frac{dy}{dx}=2x+dx
\]
The equation for a tangent to a parabola y=x^2 at a point is
LaTeX:
\[
(y-y_1)=2x_1(x-x_1)
\]
Note that we can derive some rules for the differential. Applying d to a variable that is complicated enough can be broken down into easier steps.
By the binomial theorem, for integers you can see that:
LaTeX:
\[
d(x^n)=nx^{n-1}dx
\]
The constant rule is that da is 0 when a is a constant. dπ? 0. This is clear because if f(x)=5, then it doesn't matter what the increment is, it'll be 5-5=0. Likewise d distributes over addition and subtraction. The more interesting rules to derive are the product and quotient rules
LaTeX:
\[
d(uv)=(u+du)(v+dv)-uv \\ d(uv)=uv+vdu+udv+dudv-uv \\ d(uv)=vdu+udv+dudv \\ d(uv)\approx vdu+udv
\]
LaTeX:
\[
d(\frac{u}{v})=\frac{u+du}{v+dv}-\frac{u}{v} \\ d(\frac{u}{v})=\frac{(u+du)v-u(v+dv)}{v(v+dv)} \\ d(\frac{u}{v})=\frac{uv+vdu-uv-udv}{v^2+vdv} \\ d(\frac{u}{v})=\frac{vdu-udv}{v^2+vdv} \approx \frac{vdu-udv}{v^2}
\]
LaTeX:
\[
dy=dy \\ dy=dy \frac{du}{du} \\ dy=dy \frac{du}{du}\frac{dx}{dx} \\ dy=\frac{dy}{du}\frac{du}{dx}dx \\ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\\\\ y=u^2, u=(2x+3) \\ dy=2udu=2u(2)dx=2(2x+3)(2)dx
\]
PART 3: MAXIMA AND MINIMA
Of course, once you are able to find tangents you can use them for all sorts of things! The most consistently useful of these is finding the highest and lowest values a of given curve. You just find where the differential quotient is zero and you have either a minimum or a maximum. The second derivative test can be used to find out which is which:
LaTeX:
\[
\frac{d(\frac{dy}{dx})}{dx}|_a<0 \Rightarrow y|_a=max \\\frac{d(\frac{dy}{dx})}{dx}|_a>0 \Rightarrow y|_a=min
\]
Which makes sense since the second derivative tells you if the curve is concave up or concave down by its sign. If the second derivative is also zero, such as with y=x^3 at 0, then things have gone strange and the point in question may be neither maxima nor minima.
As an aside, I'd like to complain about prevailing notation. The most common way to write the second derivative of y wrt x is:
LaTeX:
\[
\frac{d^2y}{dx^2}
\]
LaTeX:
\[
d(\frac{dy}{dx})=\frac{ddydx-dyddx}{dx^2}\\ =\frac{d^2y}{dx}-\frac{dy}{dx}\frac{d^2x}{dx}\\ \frac{d(\frac{dy}{dx})}{dx}=\frac{d^2y}{dx^2}-\frac{dy}{dx}\frac{d^2x}{dx^2}
\]
PART 4: AREAS
Unlike finding tangents, areas are easy to find... well they're easy to approximate. To find the area between a curve and the x axis between two points you just need to draw a bunch of rectangles that are very thin, and add up all the rectangles. The thinner the rectangles the better the approximation. Of course, you can't just have zero width, then you would just have a bunch of lines. In order for the dimensions to work out, the rectangles need to have a width.
Anyway, this is how the area between a curve and the x axis, and the lines x=a and x=b, is denoted in the calculus
LaTeX:
\[
\int_a^bydx
\]
The fancy capital S stands for the infinite sum of all the rectangles of hight y and width dx between x=a and x=b. If you added up all those rectangles you would get the actual area... plus or minus an infinitesimal error term that could be rounded off, just like with the derivative. Note that this gives you the signed area. If the y value is negative, then the rectangle will have a negative sign attached to it. Likewise the dx has a direction, if you swap the bounds of integration you are flipping the sign of the dx and get a negative sign infront of every rectangle, which can then be pulled out of the integral by distributivity.
But this is all just academic theory. Adding up all those rectangles of y*dx would take literally forever.
PART 4a: THE FUNDAMENTAL THEOREM OF CALCULUS
Unless there was a trick. It doesn't actually make intuitive sense for there to be a trick. Areas and Tangents don't seem like they should have anything to do with each other. But they do. It turns out that:
LaTeX:
\[
\int_a^b \frac{dy}{dx}dx
\]
LaTeX:
\[
\int_a^b \frac{dy}{dx}dx=\int_a^b1dy=y|_b-y|_a
\]
LaTeX:
\[
(y_1-y_2)+(y_2-y_3)=y_1-y_3
\]
So you're in business, so long as you can find an anti-derivative, you can find the area under a curve. The issue is that symbolic integration can sometimes be impossible using only algebraic functions. The anti-derivative of y=1/x cannot be expressed as a finite combination of additions, subtractions, multiplications, divisions, and root extractions. Even adding in the trig functions, their inverses, and the exponential and logarithmic functions only pushes the problem back. These functions are typically called the "elementary functions", and they are not closed under anti-differentiation. Still, it is useful to try and find symbolic anti-derivatives in terms of the elementary functions, because where we can find such expressions we can evaluate areas exactly.
Because mathematicians never let a symbol go to waste, the notation for anti-differentiation is exactly the same as for the integral, just without the bounds of integration.
PART 5: INTEGRALS
Where the process of differentiating can be almost mechanical, the art of finding anti-derivatives is much fuzzier. Most methods of trying to find anti-derivatives is based on reversing the differentiation rules. Like the derivative, the integral distributes over addition and subtraction.
Integration by Parts is just the product rule in reverse:
LaTeX:
\[
d(uv)=udv+vdu \\ \int d(uv)=\int (udv+ vdu) \\ uv=\int udv+\int vdu \\ \int udv=uv-\int vdu
\]
Integration by Substitution is literally just the chain rule in reverse. With Trig Substitution just being fancy with u-sub
LaTeX:
\[
\int \frac{du}{dv}dv=\int du
\]
LaTeX:
\[
\int x^ndx=\frac{x^{n+1}}{n+1}
\]
Note that due to how any constant becomes zero when differentiated, when finding anti-derivatives you need to add a constant term to the expression. Which constant? We don't know enough to say unless given more information. So it is typically just noted as +C, and left as is.
PART 6: TRANSCENDENTAL FUNCTIONS
There are a number of functions in math that "transcend" algebra. Most functions are transcendental, but the ones that actually are commonly encountered are fairly few. The Trig Functions, the Logarithmic Functions, and their inverses the arctrig functions and the exponential functions are the ones that actually were encountered before the invention of the Calculus dramatically expanded the range of functions people interacted with. So in this final part I'm going to show how to differentiate these functions from first principles... well the trig functions and the exponential and log anyway. The inverse trig functions are left as an exercise to the readers.
First is the trig functions. We only actually need one of them, if you know how to differentiate sine or cosine then the differentiation rules will let you differentiate any of the trig functions. But I'll go over both sine and cosine at the same time, because it's easy to do both at once when your working with differentials.
First we define the sine and cosine functions geometrically. Given a circle of radius r centered at the origin
LaTeX:
\[
\sin {\theta} = \frac{y}{r} \\ \cos {\theta} = \frac{x}{r}
\]
LaTeX:
\[
ds=rd\theta \\ x^2+y^2=r^2 \\ 2xdx+2ydy=0 \Rightarrow xdx+ydy=0 \\ ds^2=dx^2+dy^2
\]
LaTeX:
\[
ds^2=r^2d\theta^2=dx^2+dy^2 \\ =dx^2(1+(\frac{dy}{dx})^2)=dy^2((\frac{dx}{dy})^2+1)\\ =dx^2(1+(\frac{x}{y})^2)=dy^2((\frac{y}{x})^2+1) \\ =dx^2(\frac{y^2+x^2}{y^2})=dy^2(\frac{y^2+x^2}{x^2}) \\ r^2d\theta^2=r^2\frac{dx^2}{y^2}=r^2\frac{dy^2}{x^2} \\ d\theta^2=\frac{dx^2}{y^2}=\frac{dy^2}{x^2}\\ dx^2=(yd\theta)^2 \; , \; dy^2=(xd\theta)^2
\]
LaTeX:
\[
dx=-yd\theta \; , \; dy=xd\theta \\ \frac{dx}{r}=-\frac{y}{r}d\theta \; , \; \frac{dy}{r}=\frac{x}{r}d\theta \\ d\frac{x}{r}=-\frac{y}{r}d\theta \; , \; d\frac{y}{r}=\frac{x}{r}d\theta \\ d(\cos(\theta))= -\sin(\theta)d\theta \; , \; d(\sin(\theta))=\cos(\theta)d\theta
\]
The logarithm was invented for a practical purpose: addition is easier than multiplication, so if we had a way to turn multiplication into addition we could spend less time in tedious computation and more time doing the interesting stuff. As such a logarithmic function is any non-zero function that obeys the following relationship:
LaTeX:
\[f(xy)=f(x)+f(y)\]
LaTeX:
\[
\int_1^{ab}\frac{1}{x}dx=\int_1^{a}\frac{1}{x}dx+\int_a^{ab}\frac{1}{x}dx \\ \mathrm{let} \; x=au, u=\frac{x}{a} \\ dx=adu \\ \int_a^{ab}\frac{1}{x}dx=\int_1^{b}\frac{1}{au}adu =\int_1^{b}\frac{1}{u}du \\ \mathrm{so} \\ \int_1^{ab}\frac{1}{x}dx=\int_1^{a}\frac{1}{x}dx+\int_1^{b}\frac{1}{u}du
\]
The derivative of the inverse function is the reciprocal of the derivative. This makes sense because the derivative of y wrt x is the change in y divided by the change in x. Thus if you divide dx by dy instead it's like flipping the axes. Thus, for the exponential function of the same base, call that base "e" for now:
LaTeX:
\[
y=e^x \\ ln(y)=x \\ \frac{dy}{y}=dx \\ \frac{1}{y}=\frac{dx}{dy} \\ \frac{dy}{dx}=y=e^x \\ d(e^x)=e^xdx
\]
The differentials of the inverse trig functions can be found in the same way, which is left as an exercise to the reader.
This finally lets us prove the power rule for all real numbers, rational and irrational:
LaTeX:
\[
d(x^n)=d(e^{n \ln (x)}) \\ =e^{n \ln (x)}d(n\ln(x)) \\ =x^n(\ln(x)dn+nd(\ln(x))) \\ =x^n(0+n\frac{dx}{x}) \\ =n\frac{x^n}{x}dx\\ =nx^{n-1}dx
\]
And to round us off, here's the functional power rule:
LaTeX:
\[
d(u^v)=d(e^{v \ln(u)}) \\ =e^{v \ln(u)}d(v \ln(u)) \\ =u^v(\ln(u)dv+vd(\ln(u))) \\ =\ln(u)u^vdv+\frac{vu^v}{u}du \\ d(u^v)=\ln(u)u^vdv+vu^{v-1}du
\]
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