CLASS_11_MATHS_SOLUTIONS_NCERT

Class XI Chapter 13 – Limits and Derivatives Maths
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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
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