Class XI Chapter 13 – Limits and Derivatives Maths ______________________________________________________________________________ f x h f x f ' x lim h0 h cot x h cot x lim h0 h lim 1 cos x h cos x h0 h sin x h sin x 1 sin xcos( x h) cos xsin x h lim h0 h sin xsin x h lim 1 sin( x x h) h0 h sin x sin( x h ) 1 1 sin( h) .lim sin x h0 h sin( x h) 1 sinh 1 . lim lim sin x h0 h h0 sin x h 1 1 .1. sin x sin x 0 1 2 sin x 2 cos ec x 2 cot x' cos ec x 2 Now, let f2(x) = cosec x. Accordingly, f xh ec x h By first principle, f2x h f2x f2 ' x lim h0 h 1 lim cos ec x h cos ecx h0 h lim 1 1 1 h0 h sin x h sin x 1 sin x sin x h lim h0 h sin xsin x h 2 cos 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 ______________________________________________________________________________ 2cos x x h x x h sin 1 1 2 2 .lim sin x h 0 h sin( x h) 2 2cos x h sin h 1 1 2 2 .lim sin x h0 h sin( x h) 2 sin h x h cos 1 2 2 .lim . sin x h0 h sin( x h) 2 h 2x h sin cos 1 2 2 .lim .lim sin x h0 h h0 sin( x h) 2 2 cos x h 1 2 . 1. sin x sin( x 0) 1 cos x . sin x sin x cos ecx.cot x cos ecx ' cos ecx.cot x 3 From (1), (2), and (3), we obtain 2 f ' x cos ecx cos ec x cot x cos ecx cot x 3 2 cos cot cos ec x x ecx Question 16: 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): Solution 16: cos x Let f(x) = 1sin x cos x 1sin x Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

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