Consider f(x)=x2 and g(x)=x5. Applying the power rule, we
get f′(x)=2x and g′(x)=5x4. Now, f(x)⋅g(x)=x2⋅x5=x7, so we have (f(x)⋅g(x))′=7x6, again by
the power rule. However, f′(x)⋅g′(x)=2x⋅5x4=10x5≠7x6!
In particular, this shows that (f(x)g(x))′≠f′(x)⋅g′(x). We differentiate a product by the product
rule.
The Product
Rule ddx(f(x)⋅g(x))=f′(x)⋅g(x)+f(x)⋅g′(x)
The derivative of a product is not the product of the
derivatives.
We write, briefly, that (fg)′=f′g+fg′ -- take the derivative of
one function and leave the other alone, then add to that the
derivative of the other function and leave the first alone.
We can extend the Product Rule to the product of three
functions: (fgh)′=f′gh+fg′h+fgh′.
DO: What do you think
(fghp)′ is?
Example: Since the derivative of x2 is 2x
and the derivative of ex is ex, the derivative of x2ex is
(x2ex)′=ddx(x2)ex+x2ddx(ex)=2xex+x2ex=(x2+2x)ex.