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### Chain rule

With the **chain rule** you can determine the derivative of composite functions

for the function *f* (*x*) = *y* (*u* (*v* (*x*))).

##### Explanation

We take the composite function

f(x) =y(u(x))

For the derivative applies

But now we create a transition

You cannot just do this. As Δ*x* approaches zero Δ*u* may not become zero, as otherwise a fraction is created whose denominator is 0, and that is undefined. We will check this later. For the limit of a product applies

Because Δ*x* approaches zero Δ*u* will also approach zero or will become zero. In the first limit we replace *x *by *u*, and get

In order to control the situation Δ*u = *0 we write

So we come to the conclusion that also should apply

You can guess how this continues.

##### Example 1

We take the composite function

Substitution of *u = x*^{2} + 3 gives

Applying the chain rule, the derivative is

and finally

If we first solve the function we see

Of course we find the same answer.

##### Example 2

Using the chain rule, we calculate the derivative of *y* = 2*u*^{2}– 2 where *u* = 3*x* + 1