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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.
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