Introduction / Summary of Differentiation

Hootenanny

As the title suggests this thread is intended as a summary of differentiation, it is by no means an attempt at an exhaustive discussion of the topic. A detailed knowledge of limits is not required but is useful, it is assumed that the reader has some knowledge of limits and how to implement them. As always I welcome comments and corrections either here of via PM.

The Definition of a Derivative

The derivative ([itex]f'(x)[/itex]) of a function [itex]f(x)[/itex] with respect to [itex]x[/itex] is given by the limit;

[tex]f'(x)=\lim_{h\to0}\;\frac{f(x+h)-f(x)}{h}[/tex]

Provided the limit exists. If the limit exist at a point x0 we say that the function is differentiable at this point.

As can be seen from the plots below, as [itex]h[/itex] approaches zero, the secants of the curve tend to the tangent at a point x, this gives the gradient of the curve at point x (provided the function is differentiable at point x). Note however, that in the above limit we cannot simply set [itex]h[/itex] as zero directly as our limit would [itex]\to\pm\infty[/itex] and thus, the limit would not exist. We must therefore manipulate the limit into a form where this does not occur.


Images taken from Wikipedia

Example
Find the derivative of [itex]f(x)= x^2+2[/itex]
Solution;
[tex]f'(x)=\lim_{h\to0}\;\frac{f(x+h)-f(x)}{h}\;\;\rightarrow \lim_{h\to0}\;\frac{(x+h)^2+2-(x^2+2)}{h}[/tex]
[tex]=\lim_{h\to0}\;\frac{\not{x^2}+h^2+2hx +\not{2}-\not{x^2}-\not{2}}{h}= lim_{h\to0}\;\frac{\not{h}(h+2x)}{\not{h}}[/tex]
[tex]\therefore f'(x)=2x[/tex]

Finding the Gradient and Tangent to a Curve at a Point
We already now that the derivative of a function gives the gradient of a curve at any point that is differentiable. This allows us to find the gradient of the function at any point x where the curve is differentiable. This then allows us to find the equation of the tangent to the curve at any point x where the function is differentiable.

Example
Find the equation of the tangent to the curve with equation [itex]y=f(x)=x^2+2[/itex] at x=3.
Solution;
We know that the tangent is a straight line and therefore has an equation of the form [itex]y=mx+c[/itex]. We have already found the derivative above which gives the gradient (m) of any point x on the curve y=f(x) (provided f(x) is differentiable at x). Therefore we can say that;

[tex]m=\left. \frac{dy}{dx}\;\;\right|_{x=3}=2\cdot3=6[/tex]

We can find the corresponding y value to x=3 using the original equation [itex]y=x^2+2[/itex];
[tex]y=3^2+2=11[/tex]
We can then use this information to find the equation of the tangent at point [itex](x,y)=(3,11)[/itex];
[tex]y=mx+c \rightarrow 11=3\cdot 6 + C \Rightarrow C = -7[/tex]
[itex]\therefore[/itex] the equation of the tangent to the curve [itex]y=x^2+2[/itex] at the point [itex](x,y)=(3,11)[/itex] is [itex]y=6x-7[/itex].

Hootenanny

Rules of Differentiation

Using the definition of a derivative every time we wish to differentiate a function becomes tedious after a while. It is therefore desirable to derive a series of rules which allows us to differentiate a variety of functions without having to refer to the above definition.

1. Derivative of a Constant Function
If [itex]f[/itex] has a constant value [itex]f(x)=c[/itex], then;

[tex]f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{c-c}{h}=0[/tex]

[tex]\therefore {\color{red}\boxed{{\color{black}\frac{d}{dx}\;(c) = 0}}}[/tex]

2. Power Rule for Positive Integers
If [itex]f(x)=x^{n}[/itex] and [itex] n\in\mathbb{Z},\;n>0[/itex] then;

[tex]f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to0}\frac{(x+h)^{n}-x^{n}}{h}[/tex]
Since n is a positive integer we can use the binomial theorem to expand the parentheses;

[tex]=\lim_{h\to0}\frac{\left\{\not{x^n} + nx^{-1}\cdot h + \frac{n(n-1)}{2!}\cdot x^{n-2}h^{2} + \ldots + nxh^{n-1}+h^{n}\right\}-\not{x^{n}}}{h}[/tex]

[tex]=\lim_{h\to0}\frac{nx^{-1}\cdot h + \frac{n(n-1)}{2!}\cdot x^{n-2}h^{2} + \ldots + nxh^{n-1}+h^{n}}{h}[/tex]

One h from each term in the numerator will cancel with the h is the denominator leaving;

[tex]=\lim_{h\to0}\; nx^{-1} + \frac{n(n-1)}{2!}\cdot x^{n-2}h + \ldots + nxh^{n-2}+h^{n-1}[/tex]

As [itex]h\to0[/itex] all terms except the first drop out leaving;

[tex]f'(x)=nx^{n-1}[/tex]

[tex]\therefore {\color{red}\boxed{{\color{black}\frac{d}{dx}\;n^{n} = nx^{n-1}\;\;\; n\in\mathbb{Z},\;n>0}}}[/tex]

Note that this can be shown to be true for all integers (which will be done later).

3. Constant Coefficient Rule
If [itex]f[/itex] is a differentiable function of [itex]x[/itex] and [itex]c[/itex] is constant then;

[tex]f'(x)=\lim_{h\to0}\frac{cf(x+h)-cf(x)}{h} \stackrel{Limit\;Property}{=} c\cdot\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]

Since the RHS of the equality is the definition of the derivative we can say that;

[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\;(cf(x)) = c\cdot\frac{df}{dx}}}}[/tex]

4. The Derivative of a Sum
If we take two functions of [itex]x[/itex]; [itex]u[/itex] and [itex]v[/itex], both of which are differentiable then;

[tex]\frac{d}{dx} \left(u(x)+v(x)\right) = \lim_{h\to0}\frac{(u(x+h)+v(x+h))-(u(x)+v(x))}{h}[/tex]

We can then split this into two fractions;

[tex]=\lim_{h\to0}\left\{ \frac{u(x+h)-u(x)}{h}+\frac{v(x+h)-v(x)}{h}\right\}[/tex]

[tex]=\lim_{h\to0}\frac{u(x+h)-u(x)}{h} + \lim_{h\to0}\frac{v(x+h)-v(x)}{h}\right\} = \frac{du}{dx}+\frac{dv}{dx}[/tex]

[tex]\therefore {\color{red}\boxed{{\color{black}\frac{d}{dx}\;(u+v)=\frac{du}{dx}+\fra c{dv}{dx}}}}[/tex]

The sum rule can be shown to hold for differences (i.e. [itex](u-v)[/itex]) by combining the sum rule with the constant multiple rule and setting [itex]c=-1[/itex].

Hootenanny

5. Product Rule
Suppose we have two functions of [itex]x[/itex] - [itex]u(x)[/itex] and [itex]v(x)[/itex] both differentiable at [itex]x[/itex] - and we wish to find the derivative of their product, then from our definition of the derivative we can write;

[tex]\frac{d}{dx}\;(u\dot v) = \lim_{h\to0}\frac{u(x+h)\cdot v(x+h)-u(x)\cdot v(x)}{h}[/tex]

The next step may appear nonsensical initially but is necessary so we can evaluate the limits of [itex]u[/itex] and [itex]v[/itex] separately. The next step is to add and then subtract [itex]u(x+h)\cdot v(x)[/itex] from the numerator (the additions are highlighted in blue);

[tex]=\lim_{h\to0}\frac{u(x+h)\cdot v(x+h) {\color{blue}-u(x+h)\cdot v(x) +u(x+h)\cdot v(x)} -u(x)\cdot v(x)}{h}[/tex]
[tex]=\lim_{h\to0}\left\{ \frac{u(x+h)\cdot v(x+h) {\color{blue}-u(x+h)\cdot v(x)}}{h}+\frac{{\color{blue}u(x+h)\cdot v(x)} - u(x)\cdot v(x)}{h}\right\}[/tex]

We can now take a factor of [itex]u(x+h)[/itex] from the first ratio and a factor of [itex]v(x)[/itex] from the second ratio, giving;

[tex]=\lim_{h\to0}\left\{ u(x+h)\cdot\frac{v(x+h)-v(x)}{h}+v(x)\cdot\frac{u(x+h)-u(x)}{h}\right\} = \lim_{h\to0}\; u(x+h)\cdot \lim_{h\to0}\frac{v(x+h)-v(x)}{h}+\lim_{h\to0}\frac{u(x+h)-u(x)}{h}[/tex]

Since we stated that both [itex]u[/itex] and [itex]v[/itex] are differentiable at [itex]x[/itex] this implies that both are continuous* at [itex]x[/itex], therefore as [itex]h[/itex] approaches zero, [itex]u(x+h)[/itex] approaches [itex]u(x)[/itex]. Therefore, the above limits become what is known as the product rule;

[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\;(u\cdot v)= u\cdot\frac{dv}{dx}+v\cdot\frac{du}{dx}}}}[/tex]

Example
Find;
[tex]\frac{d}{dx}\left\{ (x^4+4)\cdot (x^2-1)\right\}[/tex]

Solution;
Let [itex]u(x):=x^4+4[/itex] and [itex]v(x):=x^2-1[/itex]. First we must find [itex]u'(x)[/itex] and [itex]v'(x)[/itex] for this we can use the power and the constant rule (I will not show all the steps here since the application of the rules are simple) which gives;

[tex]\frac{du}{dx} = 4x^3 \;\;\;\;\;\;\;\;\ \frac{dv}{dx} = 2x[/tex]

Therefore, using the product rule we obtain;

[tex]\frac{d}{dx} (x^4+4)\cdot (x^2-1) =u\frac{dv}{dx}+v\frac{du}{dx}= (x^4+4)\cdot 2x + (x^2-1)\cdot 4x^2 = 2(x^5 + 2x^4 -2x^2 + 4x)[/tex]

[tex]\hline[/tex]
*To prove this would require a discussion of limits, which I have already said I will not do here. If you wish to see a proof of this, then I will be happy to oblige via PM but a knowledge of limits is required beforehand.

Hootenanny

6. The Quotient Rule
Suppose we again have two functions of [itex]x[/itex] ([itex]u(x)[/itex] and [itex]v(x)[/itex]), both of which are differentiable at [itex]x[/itex] and [itex]v(x)\neq0[/itex]. What if we wish to find the derivative of the ratio of the two functions?

[tex]\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \lim_{h\to0}\frac{\frac{u(x+h)}{v(x+h)}-\frac{u(x)}{v(x)}}{h}[/tex]

[tex]=\lim_{h\to0}\frac{v(x)\cdot u(x+h) - u(x)\cdot v(x+h)}{h\cdot v(x+h)\cdot v(x)}[/tex]

Again, here we must take the seemingly nonsensical step of adding and subtracting [itex]u(x)\cdot v(x)[/itex] from the numerator. However, this will allow us to write two difference ratios for the derivatives of [itex]u(x)[/itex] and [itex]v(x)[/itex] (again this step is highlighted in blue);

[tex]=\lim_{h\to0}\frac{v(x)\cdot u(x+h) {\color{blue}-u(x)\cdot v(x) + u(x)\cdot v(x)} -u(x)\cdot v(x+h)}{h\cdot v(x+h)\cdot v(x)}[/tex]

[tex]=\lim_{h\to0}\frac{v(x)\cdot{\color{red}\frac{u(x+h)-u(x)}{h}}-u(x)\cdot{\color{red}\frac{v(x+h)-v(x)}{h}}}{v(x+h)\cdot v(x)}[/tex]

Notice that the two ratios highlighted in red are the derivative of the functions [itex]u(x)[/itex] and [itex]v(x)[/itex] respectively. Notice also that as [itex]h\to0[/itex], then [itex]v(x+h)\cdot v(x) \to (v(x))^2[/itex]. Therefore when we evaluate our limit we obtain;

[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\left(\frac{u}{v}\right)=\ frac{v\frac{du}{dx}-u\frac{du}{dx}}{v^2}}}}[/tex]

Example
Find;
[tex]\frac{d}{dx}\left(\frac{x^2+1}{x^2-1}\right)[/tex]

Solution;
Let [itex]u(x)=x^2+1[/itex] and [itex]v(x)=x^2-1[/itex], then,

[tex]\frac{du}{dx}=2x \;\;\;\;\;\; \frac{dv}{dx}=2x\;\;\;\;\;\; (v(x))^2 = (x^2-1)^2[/tex]

[tex]\Rightarrow \frac{d}{dx}\frac{x^2+1}{x^2-1} = \frac{2x\cdot(x^2+1) - 2x\cdot(x^2-1)}{(x^2-1)^2} = \frac{4x^3+4x}{(x^2-1)}[/tex]

Hootenanny

7. Power Rule for Negative Integers
Suppose we have [itex]x^n[/itex], where [itex]n \in \mathbb{Z},\; n<0[/itex] and [itex]n=-m[/itex], where [itex]m \in \mathbb{Z},\; n>0[/itex]. Hence we can say that;

[tex]x^n=x^{-m}=\frac{1}{x^m}[/tex]

Therefore, using the quotient rule (and positive power rule) we can say that;

[tex]\frac{d}{dx}\;x^n = \frac{d}{dx}\;\frac{1}{x^m} = \frac{x^m\cdot\frac{d}{dx}(1) - 1\cdot\frac{d}{dx}\;x^m}{(x^m)^2}[/tex]

[tex]=\frac{0-mx^{m-1}}{x^{2m}} = -mx^{-m-1} \stackrel{n=-m}{=}nx^{n-1}[/tex]

This result is identical to the power rule for positive integers, we can therefore generalise the power rule thus;

[tex]\therefore {\color{red}\boxed{{\color{black}\frac{d}{dx}\;n^{ n} = nx^{n-1}\;\;\; n\in\mathbb{Z}}}}[/tex]

I do not think an example is necessary here.

minase

Can you continue proving the Power rule for all real numbers using logarithmic differentiation.

Hootenanny

Hi Minase,
Thank you for your comments. I am planning to discuss the definition and differentiation of logarithmic and exponential functions later in the tutorial. Prior to this I will consider the chain rule, implicit differentiation and differentiation of trigonometric functions.

Hootenanny

8. The Chain Rule

Note; This proof requires knowledge of Linear Approximations and Differentials. For those who have not met linearization or differentials before, I will provide a external link to a 'rough proof' later and you may skip the following section.

[tex]\hrule[/tex]

The chain rule is used to find the derivative of a composite function (in other words a function of a function).

Provided [itex]f(x)[/itex] is differentiable at [itex]\delta[/itex], then we can show that the respective change in [itex]f[/itex] as [itex]\delta[/itex] changes to [itex]\delta+\Delta x[/itex] is given by;

[tex]\underbrace{f(x+\delta)-f(x)}_{\text{Change in }f}=\underbrace{f'(\delta)}_{\text{approximation}}+\underbrace{\epsilon \Delta x}_{\text{error}}[/tex]
Where [itex]\Delta x \to 0\Rightarrow \epsilon\to0[/itex].

Let [itex]u=g(x)[/itex] be differentiable at [itex]x[/itex] and [itex]f(u)[/itex] be differentiable at [itex]g(x)[/itex]. Also, let [itex]\Delta x, \Delta y, \Delta u[/itex] be the respective changes in [itex]x, y, x[/itex]. Then from the above equation form the following system;

[tex]\Delta u = g'(x)\Delta x + \epsilon_{0}\Delta x[/tex]
[tex]\Delta y = f'(u)\Delta u + \epsilon_{1}\Delta u[/tex]

Substituting [itex]\Delta u[/itex] into [itex]\Delta y[/itex] gives;

[tex]\Delta y = f'(u)\left\{ g'(x)\Delta x + \epsilon_{0}\Delta x\right\} + \epsilon_{1}\left\{ g'(x)\Delta x + \epsilon_{0}\Delta x\right\}[/tex]

Expanding the parenthesis;

[tex]\Delta y = f'(u)g'(x)\Delta x + f'(u)\epsilon_{0}\Delta x + \epsilon_{1}g'(x)\Delta x + \epsilon_{1}\epsilon_{0}\Delta x[/tex]

Dividing throughout by [itex]\Delta x[/itex];

[tex]\frac{\Delta y}{\Delta x} = f'(u)g'(x) + f'(u)\epsilon_{0} + \epsilon_{1}g'(x) + \epsilon_{1}\epsilon_{0}[/tex]

Since we want to find the derivative at a point we let [itex]\Delta x\to0[/itex] (recall that [itex]\epsilon\to0[/itex] as [itex]\Delta x\to0[/itex]), thus obtaining;

[tex]\lim_{\Delta x\to0}\;\left( f'(u)g'(x) + f'(u)\epsilon_{0} + \epsilon_{1}g'(x) + \epsilon_{1}\epsilon_{0}\right)[/tex]

[tex]=f'(u)g'(x) \stackrel{u=g(x)}{=} f'(g(x))\cdot g(x)[/itex]

[tex]\therefore{\color{red}\boxed{{\color{black}\frac{dy}{dx}\; f\circ g(x) = \frac{dy}{dx}\; f(g(x)) = f'(g(x))\cdot g'(x) = \frac{dy}{du}\cdot\frac{du}{dx} }}}[/tex]

[tex]{\color{blue}\hrule}[/tex]

'Rough Proof' of chain rule
from World Web Math, MIT

This 'proof' is intended for those who have not met differentials and linearization previously, so that they can see that the chain rule has some grounding. This should by no means be considered a correct formal proof.

[tex]{\color{blue}\hrule}[/tex]

Example;
Find;
[tex]\frac{d}{dx}\;(2x^2+x)^3[/tex]

Solution;
Using Leibniz's notation, let [itex] u=2x^2+x[/itex], therefore we have;

[tex]\frac{d}{dx}\;(2x^2+x)^3 = \left\{ \frac{d}{du} u^3 \right\} \cdot \left\{ \frac{d}{dx} (2x^2+x) \right\}[/tex]

[tex]=3\cdot u^2 \cdot (4x+1) = 3\cdot(2x^2+x)^2\cdot(4x+1)[/tex]

[tex]\hrule[/itex]

Note that the chain rule can be used multiple times to find a derivative.

Hootenanny

A correction in post number four. The formula should read;

[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\left (\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}}}[/tex]

As an aside it can also be expressed, perhaps more succinctly, thus;

[tex]{\color{red}\boxed{{\color{black}\frac{d}{dx}\left (\frac{u}{v}\right)=\frac{v\cdot u' - u\cdot v'}{v^2}}}}[/tex]

Hootenanny

Implicit Differentiation
Up until this point we have always considered functions which can be written explicitly in the form of [itex]y=ax^n + bx^{n-1}...[/itex], these are known an explicit functions. However, some functions or equations can not be written explicitly, instead they must be define in terms of a relationship between two variables, these are known as implicit functions. An example of an implicit function would be that of a unit circle; [itex]x^2 + y^2 =1[/itex]. On occasions it may be possible to write such equations in an explicit form, however, it is often preferable to differentiate implicit functions implicitly as they are.

When we differentiate implicitly we differentiate the implicit function with respect to the desired variable (x for example); we then treat other variables (y for example) as unknown functions of x and differentiate them accordingly using the chain rule. There is no formula to learn here (except the chain rule obviously), it is simply a technique you must practise.

Example;
Find an expression for the gradient of the unit circle at any given point (x,y)

Solution;

The equation for the unit circle in Cartesian coordinates is given by [itex]x^2 + y^2 =1[/itex] . Now, to find the gradient we must differentiate both sides of the equation with respect to x;

[tex]\frac{d}{dx}\; x^2 + y^2 = \frac{d}{dx}\;1[/tex]

Now, the derivative of x² is a simple application of the chain rule; the derivative of 1 is again a simple application of the constant rule, so that leaves us with;

[tex]2x + \frac{d}{dx}\; y^2 = 0 [/tex]

Now, since y is some (unknown) function of x; we can apply the chain rule here. (Refresh your knowledge of the chain rule if required) We therefore obtain;

[tex] 2x + y\cdot\frac{dy}{dx} = 0[/tex]

[tex]\therefore \;\;\; \frac{dy}{dx} = -\frac{2x}{y}[/tex]

Hootenanny

A typo in my previous post. The penultimate line in the example should read as follows;

[tex] 2x + {\color{red}2}\cdot y\cdot\frac{dy}{dx} = 0[/tex]

And therefore the final line will read;

[tex]\therefore \;\;\; m = \frac{dy}{dx} = -\frac{x}{y}[/tex]

Apologies for any confusion.

Schrodinger's Dog

In the example given for the quotient rule the answer should be, I hope:-

[tex]\frac{d}{dx}\left(\frac{x^2+1}{x^2-1}\right)[/tex]

[tex]\Rightarrow \frac{d}{dx}\frac{x^2+1}{x^2-1} = \frac{2x\cdot(x^2-1) - 2x\cdot(x^2+1)}{(x^2-1)^2} = -\frac{4x}{(x^2-1)^2}[/tex]

You have no idea how long I puzzled over your answer before I realised that it must be wrong.

kesh

this bit is bugging me for a few reasons.

firstly as setting h = 0 would make numerator as well as denominator zero, we've no need for the intuitive idea that the expression becomes infinity.

secondly the bolded limit might be better replaced with expression and the arrow removed, as setting to zero, rather than limiting, makes it no longer a limit and means ideas like +/- infinity are out of place

an explanation of why the differential is undefined at h = 0 and the need for a limit as a consequence of this might be better

kesh

i saw a reply to my point immediately above, but then it was deleted.

?

Hootenanny

Thank you for your incisive comments, you make a valid point. I intend, perhaps sometime after Christmas, to release a 'final' or corrected version of the tutorial in pdf format with additional examples. My intention with posting the 'initial' version on the forums would, as you have done comment on the work and suggest improvements and identify flaws. I thank you again for your useful comments.

P.S. I removed my previous post as I felt that I didn't give your post sufficient attention.

You are of course correct Schrodinger, thank you for pointing the error out. I should stop composing post so late in the evening and start using Maple to check my solutions

kesh

i thought it might be something like that. with all the fuss about 0/0 in the blogsphere i thought it might be nice to show off an area where it's dealt with usefully and rigorously, namely differentiation :)

Hootenanny

Differentiation of Trigonometric Functions
Trigonometric functions has many important applications in science, especially physics. Therefore, it is important that we are able to differentiate equations involving trigonometric functions. Here I intend to illustrate how to differentiate trigonometric functions from first principles and present a series of 'rules' which I would recommend committing to memory if you are going to be dealing with trigonometric functions on a regular basis.

1. Derivative of Sine
For no particular reason I will begin with a derivative of a sine function. From out definition of the derivative we have;

[tex]f'(x) = \lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]

[tex]f(x):=\sin(x) \Rightarrow f'(x) = \lim_{h\to0}\frac{\sin(x+h)-\sin(x)}{h}[/tex]

Applying the angle sum identity [itex]{\color{blue}\sin(A+B) = \sin(A)\cos(B)+\cos(A)\sin(B)}[/itex], we obtain;

[tex]f'(x) = \lim_{h\to0} \frac{\left( \sin(x)\cos(h)+\cos(x)\sin(h) \right)-\sin(x)}{h}[/tex]

Taking a factor of [itex]\sin(x)[/itex] out we have;

[tex]f'(x) = \lim_{h\to0} \frac{\sin(x)\left( \cos(h) - 1 \right) + \cos(x)\sin(h)}{h}[/tex]

Splitting the limit into two separate limits and rewriting the fractions;

[tex]f'(x) = \lim_{h\to0} \left( \sin(x)\cdot\frac{\cos(h) - 1}{h} \right) + \lim_{h\to0} \left( \cos(x)\cdot\frac{\sin(h)}{h} \right)[/tex]

[tex]f'(x) = \sin(x)\cdot\lim_{h\to0} \left(\frac{\cos(h) - 1}{h} \right) + \cos(x) \cdot\lim_{h\to0} \left(\frac{\sin(h)}{h} \right)[/tex]

I state without proof that the following is true. I may prove them at a later time, but for the moment they should be accepted as truths. ([itex]\theta[/itex] in radians)

[tex]{\color{blue}\boxed{\hspace{1cm}(1)\;\;\lim_{\theta\to0}\frac{\sin\thet a}{\theta}= 1 \hspace{1cm}(2)\;\;\lim_{h\to0}\frac{\cos(h)-1}{h}\hspace{1cm}}}[/tex]
Note that (2) follows directly from (1)

[tex]\therefore f'(x) = \sin(x)\cdot0 + \cos(x)\cdot1[/tex]

[tex]{\color{red}\boxed{{\color{black}f'(x) = \cos(x)}}}[/tex]

As has been alluded to previously, the derivative of a function gives the gradient of that function at any point where that function is differentiable. Therefore, the cosine function represents the rate of change of the sine function. This flash animation authored by David M. Harrison, Dept. of Physics, Univ. of Toronto illustrates this rather neatly. My thanks to EmilK for bringing this resource to my attention.