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Jail rule


With the rules we have at our disposal so far, we cannot calculate some types of derivatives.

We will now see the chain rule, a formula for the derivative of the function composed of two functions. Created by Gottfried Leibniz, thechain rule was of great importance for the advancement of differential calculus.

The formula is as follows:

It can be written as:

Another similar formula is as follows:

Calculate

We proceed as follows:

We write y = ln (x² + 1). Hoping to use the derivative of lnwe will do:

u = x² + 1
y = ln u

We calculate:

   

We use the chain rule, whose first member is the derived derivative:

 

ie multiplying the derivatives obtained in the previous step:

 

We now use the expression of u which is (x² + 1), to obtain

 

Inverse Function Derivative

The inverse of the function y(x) is the function x(y):

Next: Derivatives of Trigonometric Functions