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 = nxn-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

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




    The Power Rule and the Chain Rule

The Power Rule and the Chain Rule

When y = (x2 + 1)3, y relates to the power function of x2 + 1, this is a special case of chain rule.

If n is any real number and u = g(x) is differentiable, 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