If f(x) = 6e^x cos(x), find f '(x) and f ''(x).

Question

Find F'(x) and F''(x)

Answer 1

Given: f(x) = 6e^xcos(x)

f'(x) = 6 e^(x cos(x)) d/dx(xcos(x)) apply product rule

= 6e^(x cos(x)) [ cos(x) - x sin (x)]

= 6cos (x) e^(x cos (x)) - 6 x sin (x) e^ (x cos (x))

To find f"(x) differentiate again above answer w.r.t. x {apply product rule to each term}

f"(x) = 6 cos (x) e^(x cos(x)) [ cos(x) - x sin (x)] - 6 sin (x) e^(x cos (x))

minus (-) 6 x sin (x) e^ (x cos (x)) [ cos(x) - x sin (x)] - 6e^ (x cos (x)) [sin (x) + x cos(x)]

f"(x) = 6 cos^2 (x) e^(x cos(x)) - 12 x sin (x) cos(x) e^(x cos (x)) - 12sin (x) e^(x cos(x) + 6 x cos(x) e^(x cos (x)) Please verify the last step of simplification. It is complicated expression indeed.

I hope this helps.

Shailesh Kadakia, Expert tutor with WYZANT

Former Professor, Quantum Physics.

Answer 2

f(x) = 6e^xcosx

f'(x) = 6(e^xcosx - e^xsinx) = 6e^x(cosx - sinx)

f"(x) = 6[e^x(cosx - sinx) + e^x(-sinx - cosx)] = -12e^xsinx