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, I want to differentiate really complicated functions. As a concrete

example take a look at the function f of x equals 1 plus 2x to the fifth power. Let's

try to differentiate this function. We could approach this in a couple different

ways. First of all, I could just expand it out. Alright. So I'm just going to expand

this out 1 to the fifth is just one, plus ten x, plus 40 x squared, plus 80 x cubed,

plus 80 x to the fourth, plus 32 x to the 5th. Now, it's just a polynomial so I can

fearlessly differentiate it. So, f prime of x, by differentiate this, the

derivative of one is zero, the derivative of ten x is ten, the derivative of 40 x

squared is 80 x, and 240 x squared, 320 x cubed, and 160 x to the fourth. Of course,

if we're clever at this point, we can also see that this mess factors. So it's sort

of believable as a factor of ten here, since all of these coefficients end in a

zero. This is ten times one plus eight x plus 24 x squared plus 32 x cubed plus 16

x to the fourth. What's way less obvious, I mean not obvious at all, is that this

mess also factors. It happens to be one plus two x to the fourth power. This is

not an accident. What if we instead applied the change rule to original

problem? So let's compute the derivative to the change rule. The first step is

we're going to split up the function f into a composition of two functions, g and

h, g here, the outside function is the fifth power function, and h, the inside

function is one plus two x. So if I combine those two functions, save the

composition, I get back f. Now, I want to differentiate f and, by the chain rule,

that's the derivative of the outside, add the inside function, times the derivative

of the inside function. In this case, what is the derivative of the outside function?

The derivative of g is five x to the fourth. So I'm going to take that but if

evaluate it at h. five h of x to the fourth multiply by the derivative of h.

What is the derivative of h? Well, it's two. Well, look what I got here. I've got

five, h of x is one plus two x to the fourth times two, that's ten times one

plus two x to the fourth, that's exactly what we calculated before. It's really

nice example, because it shows that we're doing the same calculation. We're

calculating derivative of the function one plus two x to the fifth power, but we're

doing it in two different ways, nevertheless, we get the same answer.

Somehow, mathematics is conspiring to be consistent. Okay, well, let's try another

example. Well, here's a more complicated function, f of x equals the square root of

x squared plus 0.0001. What's the derivative of f? We can't simply expand

this function out, and in fact, if you graph the function, you might think that

the function is not differential, because the graph of the function has this sharp

corner at the origin, but let's zoom in and see what this actually looks like if

we zoom in close enough. If we zoom in close enough, the thing doesn't look like

it has a sharp corner anymore. It actually looks like it's curved and if we zoom in

any further, the thing would look more and more like a straight line. What we're

really seeing is the function is differentiable. Now, we can verify this

algebraically, we can use our derivative laws, like a change rule, to actually

calculate the derivative of this function. We'll differentiate this by using the

change rule since this is really a composition of two functions. This is a

composition of the square root function and this polynomial, x squared plus

0.0001. Alright, so the derivative of f is the derivative of the outside function,

which is the derivative of the square root, which is 1 over 2 square root And

it's the derivative of the outside function evaluated at the inside, which is

x squared plus 0.0001. I have to multiply by the derivative of the inside function.

What is the derivative of x squared plus 0.0001? Well, that's the derivative of the

x squared, since it's a constant and the derivative of x squared is two x. So the

dirivitave of f is one over two t imes the square root of x squared plus 0.0001 times

two x. I could make that a little bit nicer looking. I could cancel these twos

and write this as x over the square root of x squared plus 0.0001. What happens at

zero? So let compute the derivative at zero. Well, if I plug in zero for x, I've

got zero over zero squared plus 0.0001. The denominator is not zero, the numerator

is zero, the derivative at zero is zero and you can see that from the graph. If I

look at when x equals zero, the tangent line at that point is horizontal, the

slope of that tangent line is zero. The derivative at zero is zero, and there's

more awesome things that you can see by looking at the derivative. If you look at,

say, the limit of the derivative as x approaches infinity, that's the limit of

this quantity, which is one, and the limit of the derivative as x approaches minus

infinity is negative one and you can see this visibly on the graph of the function.

If you plug in a really big number and look at the tangent line there, that

tangent line has slope close to one. And, if you plug in areally negative number and

look at the tangent line there, That tangent line has slope close to minus one.

Our derivative rules are revealing facts that are hidden. This function looks like

it's got a sharp corner, but we know, by applying our differentiation rules, by

using the change rule, that this function is in fact differentiable. And we know

that if we zoom in close enough, the thing looks like a straight line, the derivative

rules really revealing this structure at very small scales.