Question

The derivative of $f(\tan x)$ with respect to $g(\sec x)$ at $x = \dfrac{\pi }{4}$ where $f'(1) = 2$ and $g'(\sqrt 2 ) = 4$ is
A) $\dfrac{1}{{\sqrt 2 }}$
B) $\sqrt 2$
C) $1$
D) None of these

Answer 1

In the given question we are asked to find the derivative of a function with respect to another function. Do not misinterpret this type of question with the one in which we are asked to find the derivative of a function with respect to the given variable. Find the respective derivatives of the given functions with respect to the given variable separately.
Here we will use the concept of finding the derivative of a function with respect to another function.
Let $u = f(x)$ and $v = g(x)$ be two functions of $x$. Then to find the derivative of $f(x)$ with respect to $g(x)$ that is to find $\dfrac{{du}}{{dv}}$ we will make use of the following formula:
$\dfrac{{du}}{{dv}} = \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}}$
Thus to find the derivative of a function $f(x)$ with respect to another function $g(x)$ we differentiate both with respect to $x$ and then divide the derivative of the function$f(x)$ with respect to $x$ by the derivative of the function $g(x)$ with respect to $x$.

Complete step by step answer:
Let $y = f(\tan x)$ and $z = g(\sec x)$
Differentiating both the functions with respect to $x$ we get ,
$\dfrac{{dy}}{{dx}} = f'(\tan x){\sec ^2}x$ and $\dfrac{{dz}}{{dx}} = g'(\sec x)\sec x\tan x$
Therefore we get $\dfrac{{dy}}{{dz}} = \dfrac{{\dfrac{{dy}}{{dx}}}}{{\dfrac{{dz}}{{dx}}}} = \dfrac{{f'(\tan x){{\sec }^2}x}}{{g'(\sec x)\sec x\tan x}}$
On simplifying the above equation we get ,
$\dfrac{{dy}}{{dz}} = \dfrac{{f'(\tan x)\sec x}}{{g'(\sec x)\tan x}}$
Now putting the value $x = \dfrac{\pi }{4}$ we get ,
$\dfrac{{dy}}{{dz}} = \dfrac{{f'\left( {\tan \dfrac{\pi }{4}} \right)\sec \dfrac{\pi }{4}}}{{g'\left( {\sec \dfrac{\pi }{4}} \right)\tan \dfrac{\pi }{4}}}$
Which simplifies to
$\dfrac{{dy}}{{dz}} = \dfrac{{f'(1){{\left( {\sqrt 2 } \right)}^2}}}{{g'(\sqrt 2 )\left( {\sqrt 2 } \right)}}$
Hence on solving we get $\dfrac{{dy}}{{dz}} = \dfrac{2 \times 2}{4 \times{\sqrt 2 }}= \dfrac{1}{{\sqrt 2 }}$

So, the correct answer is “Option A”.

Note: Find the respective derivatives of the given functions with respect to the given variable separately. The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. Do not forget to find the particular solution at the end of the general solution.