If y = log (sin x), prove that: cos x cose³ x.

Basic idea:

√Second order derivative is nothing but derivative of derivative i.e.

√The idea of chain rule of differentiation: If f is any real-valued function which is the composition of two functions u and v, i.e. f = v(u(x)). For the sake of simplicity just assume t = u(x)

Then f = v(t). By chain rule, we can write the derivative of f w.r.t to x as:

√Product rule of differentiation-

√Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..

Let’s solve now:

As we have to prove: cos x cose³ x

We notice a third order derivative in the expression to be proved so first take the step to find the third order derivative.

Given, y = log (sin x)

Let’s find –

As

So lets first find dy/dx and differentiate it again.

∴

differentiating using the chain rule,

let, t = sin x and y = log t

∵ [using chain rule]

[∵ = & ]

Differentiating again with respect to x :

[ ∵ ]

Differentiating again with respect to x:

using the chain rule and

[ ∵ cot x = cos x/sin x]