Question

The differential coefficient of f (sin x) with respect to x where f (x) = log x is,
$\left( a \right)$ tan x
$\left( b \right)$ cot x
$\left( c \right)$ f (cos x)
$\left( d \right)\dfrac{1}{x}$

Answer 1

In this particular question substitute the asked differential coefficient in place of x in the given function then use the concept that the differentiation of $\dfrac{d}{{dx}}\log \left( {g\left( x \right)} \right) = \dfrac{1}{{g\left( x \right)}}\dfrac{d}{{dx}}g\left( x \right)$ so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given function:
f(x) = log x..................... (1)
Now we have to find the differential coefficient of f (sin x) with respect to x.
So first substitute sin x in place of x in the first equation we have,
Therefore, f (sin x) = log (sin x)
Now differentiation this equation w.r.t x we have,
$\Rightarrow \dfrac{d}{{dx}}f\left( {\sin x} \right) = f'\left( {\sin x} \right) = \dfrac{d}{{dx}}\left( {\log \left( {\sin x} \right)} \right)$
Now as we know that the differentiation of $\dfrac{d}{{dx}}\log \left( {g\left( x \right)} \right) = \dfrac{1}{{g\left( x \right)}}\dfrac{d}{{dx}}g\left( x \right)$ so use this property in the above equation we have,
$\Rightarrow \dfrac{d}{{dx}}f\left( {\sin x} \right) = f'\left( {\sin x} \right) = \dfrac{1}{{\sin x}}\left( {\dfrac{d}{{dx}}\sin x} \right)$
Now as we know that the differentiation of sin x is cos x, so use this property in the above equation we have,
$\Rightarrow \dfrac{d}{{dx}}f\left( {\sin x} \right) = f'\left( {\sin x} \right) = \dfrac{1}{{\sin x}}\cos x$
Now as we know that (cos x/sin x) = cot x so we have,
$\Rightarrow \dfrac{d}{{dx}}f\left( {\sin x} \right) = f'\left( {\sin x} \right) = \cot x$
So the differential coefficient of f (sin x) is cot x.
So this is the required answer.
Hence option (B) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the basic standard differentiation formula which is the basis of the above problem and which is all stated above and also recall the basic trigonometric properties such that (cos x/sin x) = cot x.