How do you differentiate f(x)=e^x*sin^2x using the product rule?
1 Answer
Aug 16, 2017
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)" ")|)