Question

Find the second derivative of:

(i) x sin x

(ii) e^2x cos 3x

(iii) x³ log x

Answer 1

(i) x sin x

Differentiating with respect to x

f’(x)= sinx + x cos x

Differentiating with respect to x

f’’(x) = cos x +cos x – x sin x = -sin x + 2 cos x

(ii) e^2x cos 3x

Differentiating with respect to x

f’(x) = 2 e^2xcos 3x + e^2x(-sin 3x).3

= 2e^2x cos 3x – 3e^2x sin 3x

Differentiating with respect to x

f’’(x) = 2.2e^2xcos 3x + 2e^2x(-sin 3x).3 – 3.2e^2xsin3x – 3e^2xcos 3x.3

= 4e^2xcos3x - 6e^2xsin 3x – 6e^2xsin3 x – 9e^2xcos 3x

=-12e^2xsin 3x – 5e^2xcos3x

(iii) x³ log x

Differentiating with respect to x

Differentiating with respect to x

= 6x log x + 3x +2x

= 6x log x +5x