The Product Rule For Derivatives

The Product Rule for derivatives is a way to find the derivative of a function H(x) that is the prodcut of two functions f(x) and g(x).  So, if H(x)=f(x)g(x), then, the product rule says that the derivative of H, H'(x) is as follows:

          
H'(x)=f'(x)g(x)+f(x)g'(x)

That is, the product of the derivative of the first function, times the second plus the first function times the derivative of the second function.

Example:

If we have a function H(x)=e^xsin(x), then, we can easily identify the two functions as f(x)=e^x and g(x)=sin(x), then, applying the product rule, we see that:

H'(x)=e^xsin(x)+e^xcos(x)=e^x[sin(x)+cos(x)]

Note here that the derivative of e^x is e^x itself and the derivative of sin(x) is cos(x).