Find d^2 (y) / dx^2 and d^3 (y) / dx^3 for y = (e^x)sin(x)

Answer 1

Just use the product rule twice to find the second derivative and 3 times to find the third derivative.

The product rule: where f(x) = u•v then f' = u'•v + u•v' and u = e^x and v = sin(x) so u' = e^x and v' = cos(x) meaning f' = e^xsin(x) + e^xcos(x), Do it again to find the second derivative:

f'' = [e^xsin(x) + e^xcos(x)] + [e^xcos(x) - e^xsin(x)] = 2e^xcos(x)

f''' = 2e^xcos(x) - 2e^xsin(x) = 2e^x[cos(x) - sin(x)]