Question

If y = log (sin x), find $\dfrac{{dy}}{{dx}}$.

Answer 1

Hint:
Here the given function is composition of two functions, which is commonly known as composite function. Also, It is composed of $log x$ and $\sin x$. To differentiate such functions we need to use chain rule i.e. (f(g(x)))’ = f'(g(x))⋅g'(x)

Complete step by step solution:
This question is based on the chain rule of differentiation chapter. Where the chain rule differentiation is that The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate the composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x) = x²..The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.
Given that,
$y = log (sin x)$
Now,
On differentiating, w ⋅ r ⋅ t ⋅ x, we get
⇒ $\dfrac{{dy}}{{dx}} = \dfrac{{1}}{{\sin x}} \dfrac{d}{dx}{\sin x}$
⇒ $\dfrac{{dy}}{{dx}} = \dfrac{{\cos x}}{{\sin x}}$ -----(A)
We know that,
 $\dfrac{{\cos x}}{{\sin x}} = \cot x$
From equation (A)

⇒ $\dfrac{{dy}}{{dx}} = \cot x$

Note:
This question is based on chain rule differentiation. The chain rule of differentiation is The chain rule states that the derivative of f(g(x)) is f'(g(x))⋅g'(x). In other words, it helps us differentiate *composite functions*. For example, sin(x²) is a composite function because it can be constructed as f(g(x)) for f(x)=sin(x) and g(x)=x². The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus
In this type of question differentiate w ⋅ r ⋅ t ⋅ x, $\dfrac{{dy}}{{dx}} = \dfrac{{\cos x}}{{\sin x}}$.