Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ d a d a d f ' x cos x 4 2 dx x dx x dx d 4 d 2 d a x b x cos x dx dx dx 5 3 d n n1 d a4x b2x sin x x nx and cos x sin x dx dx 4a 2b sin x 5 3 x x Question **11**: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non zero constants and m and n are integers): 4 x 2 Solution **11**: Let f(x) = 4 x 2 d d d f ' x 4 x 2 4 x 2 dx dx dx 1 1 d 1 1 2 2 4 x 0 4 x dx 2 1 2 2 2x x Question 12: 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): (ax + b) n Solution 12: n n ax b Accordingly f x h ax hb ax ah b Let f x . , { } By first principle, f x h f x f ' x lim h0 h lim h0 ax ah bax b h n n 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 ______________________________________________________________________________ lim h0 n ax b 1 ax b n ah ax b h n ah 1 1 n ax b ax b lim h0 h 2 n 1 ah n( n 1 ah ax b lim 1 n ... 1 h0 n ax b 2 ax b (using binomial theorem) n n ax b na ax b na ax b 1 ah nn1 2 b n 2 2 1 n a h = ax b lim n ... Terms containing higher degrees of h h0 2 n ax b ax 2 2 n na n( n 1) a h ax b lim ... h 0 2 ax b 2ax b n na ax b 0 ax b Question 13: 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): (ax + b) n (cx + d) m Solution 13: Let f(x) = (ax + b) n (cx + d) m By Leibnitz product rule, n d m m d n f ' x ax b cx d cx d ax b 1 dx dx 1 m , 1 m 1 1 ' x lim Now let f x cx d f x h cx ch d f 1 h h 0 f x h f x Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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Class XI Chapter 12 - Introduction

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