Find the derivative of the following function (where n is an integer): sinⁿx

Solution:

The given function is f(x) = sinⁿx.

It can be written as 

f(x) = sin x sin x sin x ... sin x (n times).

Using the product rule, its derivative is,

d/dx (f(x)) = [ sin x sin x ... sin x (n-1) times d/dx (sin x)
+ sin x sin x ... sin x (n-1) times d/dx (sin x)
+ ... + sin x sin x ... sin x (n-1) times d/dx (sin x) ] (n times)

= sin x sin x ... sin x (n-1) times [ cos x + cos x + ... + cos x (n times)]

= sin^n-1 x [n cos x]

= n sin^n-1x cos x 

NCERT Solutions Class 11 Maths Chapter 13 Exercise ME Question 19

Find the derivative of the following function (where n is an integer): sinⁿx

Summary:

The derivative of the given function sinⁿx is n sin^n-1x cos x