# Find the second derivative of:(i) x sin x(ii) e2x cos 3x(iii) x3 log x

Differentiating with respect to x

f’(x)= sinx + xcosx

Differentiating with respect to x

f’’(x) = cosx +cosx – xsinx

= -sinx + 2cosx

(ii) e2x cos 3x

Differentiating with respect to x

f’(x) = 2e2xcos3x + e2x(-sin3x).3

= 2e2xcos3x – 3e2xsin3x

Differentiating with respect to x

f’’(x) = 2.2e2xcos3x + 2e2x(-sin3x).3 – 3.2e2xsin3x – 3e2xcos3x.3

= 4e2xcos3x - 6e2xsin3x – 6e2xsin3x – 9e2xcos3x

=-12e2xsin3x – 5e2xcos3x

(iii) x3 log x

Differentiating with respect to x

f’(x) =

f’(x) = 3x2log x + x2

Differentiating with respect to x

= 6x log x + 3x +2x

= 6x log x +5x

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