Q. 23.5( 2 Votes )

Find the second derivative of:

(i) x sin x

(ii) e2x cos 3x

(iii) x3 log x

Answer :

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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