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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))).



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 Δx approaches zero Δu 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 Δx approaches zero Δu 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 Δu = 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 = x2 + 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 = 2u2– 2 where u = 3x + 1


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