Question

If $y = {[{(\tan x)^{\tan x}}]^{\tan x}}$, then what is the value of $\dfrac{{dy}}{{dx}}$ at $x = \dfrac{\pi }{4}$ ?
A. $2$
B. $\dfrac{1}{2}$
C. $1$
D. $0$

Answer 1

Hint: In this question, we are given an equation whose differentiated value is to be found at a given value of $x$. The given equation is a complex equation. Hence, it is to be solved by taking log on both sides. After taking log, we will solve and simplify the equation using some of the properties of log. Once the equation has been simplified, differentiate the equation with respect to $x$. Then put the given value of x and find $\dfrac{{dy}}{{dx}}$ in numeric terms.

Formula used: 1) $\log \left( {{a^b}} \right) = b\log a$
2) $\tan \dfrac{\pi }{4} = 1$
3) $\sec \dfrac{\pi }{4} = \sqrt 2 $

Complete step-by-step answer:
We are given that $y = {[{(\tan x)^{\tan x}}]^{\tan x}}$ at $x = \dfrac{\pi }{4}$.
Such complicated equations which are in the form of ${\left( {{\text{function}}} \right)^{{\text{function}}}}$ are solved by taking log on both sides. This is because log simplifies the question.
$ \Rightarrow y = {[{(\tan x)^{\tan x}}]^{\tan x}}$
Taking log on both sides,
$ \Rightarrow \log y = \log {[{(\tan x)^{\tan x}}]^{\tan x}}$
Simplifying using the property- $\log \left( {{a^b}} \right) = b\log a$,
$ \Rightarrow \log y = \tan x\log [{(\tan x)^{\tan x}}]$
Using the same property once more,
$ \Rightarrow \log y = {\tan ^2}x\log \left( {\tan x} \right)$
Differentiate with respect to x and use product rule,
$ \Rightarrow \dfrac{{d(\log y)}}{{dx}} = {\tan ^2}x\dfrac{{d[\log (\tan x)]}}{{dx}} + \log (\tan x)\dfrac{{d({{\tan }^2}x)}}{{dx}}$
Simplifying further,
$ \Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \dfrac{{{{\tan }^2}x}}{{\tan x}}({\sec ^2}x) + 2\log (\tan x)\tan x{\sec ^2}x$
$ \Rightarrow \dfrac{1}{y}\dfrac{{dy}}{{dx}} = \tan x{\sec ^2}x + 2\log (\tan x)\tan x{\sec ^2}x$
Shifting $\dfrac{1}{y}$ to RHS and putting $y = {[{(\tan x)^{\tan x}}]^{\tan x}}$,$ \Rightarrow \dfrac{{dy}}{{dx}} = {[{(\tan x)^{\tan x}}]^{\tan x}}[\tan x{\sec ^2}x + 2\log (\tan x)\tan x{\sec ^2}x]$
Putting $x = \dfrac{\pi }{4}$,
$ \Rightarrow \dfrac{{dy}}{{dx}} = {[{(\tan \dfrac{\pi }{4})^{\tan \dfrac{\pi }{4}}}]^{\tan \dfrac{\pi }{4}}}[\tan \dfrac{\pi }{4}{\sec ^2}\dfrac{\pi }{4} + 2\log (\tan \dfrac{\pi }{4})\tan \dfrac{\pi }{4}{\sec ^2}\dfrac{\pi }{4}]$
We know that the value of $\tan \dfrac{\pi }{4} = 1$ , and ${\sec ^2}\dfrac{\pi }{4} = 2$.
Substituting these values in the above equation and solving,
$ \Rightarrow 1[2 + 0]$
$ \Rightarrow 2$

Hence, the answer is option (A).

Note: Logarithm is the inverse of exponent. Some of the basic logarithmic functions are ${\log _a}x + {\log _a}y = {\log _a}xy$
ii) ${\log _a}x - {\log _a}y = {\log _a}\dfrac{x}{y}$
iii) ${\log _a}b = \dfrac{{{{\log }_c}b}}{{{{\log }_c}a}}$
These are the most commonly used logarithmic expressions.