Find the derivative of the following functions (it is to be understood that n is an integer): x / sinⁿx

Solution:

Finding the derivative of sinⁿx:

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 

Finding the derivative of the given function:

The given function is g(x) = x / sinⁿx.

Its derivative using the quotient rule is,

d/dx (g(x)) = [ (sinⁿx) d/dx (x) - (x) d/dx (sinⁿx) ] / (sinⁿx)²

= [ (sinⁿx) (1) - x (n sin^n-1x cosx) ] / (sin²ⁿx) (using derivative formulas)

= [ sin^n-1x (sin x - nx cos x) ] / (sin²ⁿx)

= (sin x - nx cos x) / (sin^2n-n+1x)   (Using a property of exponents a^m/aⁿ = 1/a^n-m]

= (sin x - nx cos x) / (sinⁿ⁺¹x)

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NCERT Solutions Class 11 Maths Chapter 13 Exercise ME Question 30

Find the derivative of the following functions (it is to be understood that n is an integer): x / sinⁿx

Summary:

The derivative of the given function x / sinⁿx is (sin x - nx cos x) / (sinⁿ⁺¹x)