2.2, The Product and Quotient Rules

Noticeably absent from our small collection of algebra rules about computing derivatives is a rule for computing derivatives of products or quotients.  This is because those rules are a bit more complicated than one might first expect.

The product rule

The derivative of a function which can be written as a product, such as f(x) = x^3sin x  is not the product of the derivatives, but a sum involving derivatives of the factors.

If

f(x) = u(x)v(x)

then

f'(x) = u'(x)v(x)+u(x)v'(x)

So the derivative of  f(x) = x^3sin x  is  f'(x) = 3x^2 sin x + x^3 cos x.

The quotient rule

If

f(x) = u(x)/v(x)

then

f'(x) = [u'(x)v(x)-u(x)v'(x)]/v^2(x)

For example, the derivative of  f(x) = x^3/sin x  is  f'(x) = [3x^2 sin x - x^3 cos x]/sin^2 x.