The Product Rule
If we know how to find the derivatives of two functions, f(x)
and g(x), can we then find the derivative of the product function
f(x) g(x) ? As you will see below, the derivative of
f(x) g(x) is not simply f'(x) g'(x). However,
there is a formula that allows you to correctly find the derivative of
f(x) g(x), and this formula involves f(x), g(x), f'(x)
and g'(x).
The mathematics below shows two specific functions f(x) and g(x).
As you change the definitions of one or both of these functions, you will
see the derivatives change automatically. The goal of this exploration
is to find a symbolic expression for the derivative of f(x) g(x)
that works in all cases.