How do you differentiate #f(x)=e^x*sin^2x# using the product rule?
1 Answer
Aug 16, 2017
Answer:
Explanation:
We're asked to find the derivative
#d/(dx) [e^x*sin^2x]#
Using the power rule*, which is
#d/(dx) [uv] = v(du)/(dx) + u(dv)/(dx)#
where

#u = e^x# 
#v = sin^2x# :
#f'(x) = sin^2xd/(dx)[e^x] + e^xd/(dx)[sin^2x]#
The derivative of
#f'(x) = e^xsin^2x + e^xd/(dx)[sin^2x]#
To differentiate the
#d/(dx) [sin^2x] = d/(du) [u^2] (du)/(dx)#
where

#u = sinx# 
#d/(du) [u^2] = 2u# (from power rule):
#f'(x) = e^xsin^2x + e^x*2sinxd/(dx)[sinx]#
The derivative of
#f'(x) = e^xsin^2x + 2e^xsinxcosx#
Or
#color(blue)(ulbar(stackrel(" ")(" "f'(x) = e^xsinx(sinx + 2cosx)" "))#