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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 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.

 


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