Question

How do you differentiate $f(x) = \tan (\ln x)$ using the chain rule?

Answer 1

This question is from the topic of differentiation. In this question we need to find the derivative of function $f(x) = \tan (\ln x)$ using the chain rule. To solve this question we need to know the conditions of chain rule of differentiation and the derivative of functions $\tan x$ and $\ln x$.

Complete step by step answer:
Let us try to solve this question in which we are asked to find the derivative of function $f(x) = \tan (\ln x)$. Before differentiating this, let’s have a look at definition of chain rule, suppose a function $f(x) = g(h(x))$ such that both $g$ and $h$ are differentiable with respect to $x$ then $f$ is also differentiable and its differentiation is given by $f'(x) = g'(h(x)) \cdot h'(x)$ where $f'(x) = \dfrac{{d(f(x))}}{{dx}}$ and similarly $g'$ and $h'$ are derivatives of functions $g$ and $h$respectively.
Now, let’s find the derivative of function$f(x) = \tan (\ln x)$. Function to derive $f(x) = \tan (\ln x)$ is composition of differentiable functions $\ln (x)$ and$\tan x$. So for the derivative of the function $f(x) = \tan (\ln x)$ we can use the chain rule of differentiate. After applying chain rule to function $f(x) = \tan (\ln x)$, we get
$\dfrac{{d(\tan (\ln x))}}{{dx}} = \dfrac{{d(\tan (\ln x))}}{{dx}} \cdot \dfrac{{d(\ln x)}}{{dx}}$ $eq(1)$
As we know that $\dfrac{{d(\tan x)}}{{dx}} = {\sec ^2}x$. So we have,
$\dfrac{{d(\tan (\ln x))}}{{dx}} = {\sec ^2}(\ln x)$ $eq(2)$
And, also we know that $\dfrac{{d(\ln x)}}{{dx}} = \dfrac{1}{x}$. So we have,
$\dfrac{{d(\ln x)}}{{dx}} = \dfrac{1}{x}$ $eq(3)$
Now, putting back the value of $eq(2)$ and $eq(3)$in$eq(1)$, we get the derivative of$\ln (5x)$.Hence the derivative of function
$\dfrac{{d(\tan (\ln x))}}{{dx}} = {\sec ^2}(\ln x) \cdot \dfrac{1}{x} \\\ \therefore\dfrac{{d(\tan (\ln x))}}{{dx}}= \dfrac{{{{\sec }^2}(\ln x)}}{x}$

Note: To solve these types of questions in which we are asked to find the derivative of a given function. For solving this type of question we are required to have knowledge of how to find derivatives of a function, differentiability of common function and properties of differentiation such sum rule, product rule, division rule and chain rule.