How do you differentiate #f(x)=e^(-x)sinx# using the product rule?

1 Answer
Dec 14, 2015

Answer:

#f'(x) = e^(-x) (cos x - sin x)#

Explanation:

The product rule is as follows:
if your function is a product of two functions,

#f(x) = g(x) * h(x)#

then the derivative is

#f'(x) = g'(x) * h(x) + g(x) * h'(x)#

In your case, #g(x) = e^(-x)# and #h(x) = sin x#.

Let's find the derivatives #g'(x)# and #h'(x)#:

#g(x) = e^(-x) color(white)(xxx) => color(white)(x) g'(x) = -e^(-x)#

#h(x) = sin x color(white)(xxi) => color(white)(x) h'(x) = cos x#

So, in total, your derivative is:

#f'(x) = g'(x) * h(x) + g(x) * h'(x)#

#color(white)(xxxx) = -e^(-x) sin x + e^(-x) cos x#

#color(white)(xxxx) = color(white)(x) e^(-x) (cos x - sin x)#