Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ 2sin x sec x cos x 2 sec x 1 2sec xtan x 2 sec x 1 Question 19: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed nonzero constants and m and n are integers): sin n x Solution 19: Let y = sin n x. Accordingly, for n = 1, y = sin x. dy d cos x, i. e., sin x cos x dx dx For n = 2, y = sin 2 x. dy d sin xsin x dx dx sin x 'sin x sin x sin x ' [By Leibnitz product rule] =cosxsinx+sinxcosx =2sinxcosx 1 For n = 3, y = sin 3 x. dy d 2 sin xsin x dx dx 2 2 sin x 'sin x sin x sin x ' [By Leibnitz product rule] 2 =cosxsin x+sinx 2sinx cosx [Using 1 ] 2 2 =cosxsin 2sin cos x x x 2 3sin xcos x d n n1 We assert that sin x nsin xcos x dx Let our assertion be true for n = k. d k k1 i.e., sin x ksin xcos x ….(2) dx Consider d k1 d k sin x sin xsin x dx dx Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ k k x x x x sin 'sin sin sin ' [By Leibnitz product rule] k-1 k =cosxsin x+sinx ksin cosx [Using 2 ] k k =cosxsin 2sin cos x x x k k 1 sin xcos x Thus, our assertion is true for n = k + 1. d n d n1 Hence, by mathematical induction, sin x nsin xcos x dx dx Question 20: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s absin x are fixed non-zero constants and m and n are integers): cd cos x Solution 20: absin x Let f(x) = cd cos x By quotient rule, f c d cos x a bsin xa bsin x c d cos x ' x dx dx 2 c d cos x c d cos xbcos xa bsin xd sin x 2 c d cos x 2 2 cbcos x bd cos x ad sin x bd sin x 2 c d cos x 2 2 bc cos x ad sin x bd cos x sin x d c d cos x bc cos x ad sin x bd 2 c d cos x 2 d Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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