Derivative Formula

In previous sections, various formulas have been introduced to calculate derivatives. However, it is not possible to have a formula for every possible situation. Thus, basic formulas like d(xⁿ)/dx = nx^n-1 need to be generalized so they can be used in a variety of cases.

Suppose y = (x² + 1)³, how can dy/dx be calculated using derivative formula?

Let f(u) = u³, where u = g(x) = x² + 1. Then

The derivative of f and g can be calculated according to derivative formula. A rule is needed to calculate the derivative of F which is a composite function. This rule is named as Chain Rule.

Chain Rule

The Chain Rule states:

If the derivatives of g(x) and f(g(x)) exist, and F = f g is the composite function defined by F(x) = f(g(x)), then the derivative of F(x) exists and is given by

     F^'(x) = f ^'(g(x))g^'(x)

In other words, If both y = f(u) and u = g(x) are differentiable functions, then

Returning to the example y = (x² + 1)³, dy/dx can be easily be determined.

Let y = f(u) = u³ and u = g(x) = x² + 1, then

so