Question

What is the derivative of \[{{\tan }^{3}}\left( {{x}^{4}} \right)\] ?

Answer 1

Hint: In this type of question students have to use the basic concept of derivatives. The given function is a trigonometric function of a function of x. So students have to apply the formula of derivative of a function of a function. Also they have to use the derivative of \[\tan x\] and derivative of \[{{x}^{n}}\] . Hence, the student should use \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\], \[\dfrac{d}{dx}\left( \tan x \right)={{\sec }^{2}}x\] and \[\dfrac{d}{dx}\left( f\left( g\left( x \right) \right) \right)=f'\left( g\left( x \right) \right)\dfrac{d}{dx}\left( g\left( x \right) \right)\] . Finally rearrange all the terms and write your result.

Complete step-by-step answer:
Now, here we have to find out derivative of \[{{\tan }^{3}}\left( {{x}^{4}} \right)\]so consider, \[\dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)\]
By Applying, the derivative of function of a function that is \[\dfrac{d}{dx}\left( f\left( g\left( x \right) \right) \right)=f'\left( g\left( x \right) \right)\dfrac{d}{dx}\left( g\left( x \right) \right)\] and \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\] We get,
\[\Rightarrow \dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=3{{\tan }^{2}}\left( {{x}^{4}} \right)\dfrac{d}{dx}\left( \tan \left( {{x}^{4}} \right) \right)\]
Now, again we apply \[\dfrac{d}{dx}\left( f\left( g\left( x \right) \right) \right)=f'\left( g\left( x \right) \right)\dfrac{d}{dx}\left( g\left( x \right) \right)\] along with \[\dfrac{d}{dx}\left( \tan x \right)={{\sec }^{2}}x\] so we get,
\[\Rightarrow \dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=3{{\tan }^{2}}\left( {{x}^{4}} \right)\left[ {{\sec }^{2}}\left( {{x}^{4}} \right)\dfrac{d}{dx}\left( {{x}^{4}} \right) \right]\]
Finally by using \[\dfrac{d}{dx}\left( {{x}^{n}} \right)=n{{x}^{n-1}}\] we can write,
\[\Rightarrow \dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=3{{\tan }^{2}}\left( {{x}^{4}} \right)\left[ {{\sec }^{2}}\left( {{x}^{4}} \right)4{{x}^{3}} \right]\]
Now, combine the constants so we get,
\[\Rightarrow \dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=12{{\tan }^{2}}\left( {{x}^{4}} \right)\left[ {{\sec }^{2}}\left( {{x}^{4}} \right){{x}^{3}} \right]\]
By rearranging the terms present we can write,
\[\Rightarrow \dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=12{{x}^{3}}{{\tan }^{2}}\left( {{x}^{4}} \right){{\sec }^{2}}\left( {{x}^{4}} \right)\]
This is the final answer of the derivative.
Hence, the derivative of \[{{\tan }^{3}}\left( {{x}^{4}} \right)\] is \[12{{x}^{3}}{{\tan }^{2}}\left( {{x}^{4}} \right){{\sec }^{2}}\left( {{x}^{4}} \right)\]

Note: In this type of question student may make mistake when they find the derivative of \[\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)\] one may write \[\dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=3{{\tan }^{2}}\left( {{x}^{4}} \right)\dfrac{d}{dx}\left( {{x}^{4}} \right)\] and forgot to take derivative of \[\tan \left( {{x}^{4}} \right)\] . Also student may find the derivative of \[\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)\] and they write the final answer as \[\dfrac{d}{dx}\left( {{\tan }^{3}}\left( {{x}^{4}} \right) \right)=3{{\tan }^{2}}\left( {{x}^{4}} \right)\left[ {{\sec }^{2}}\left( {{x}^{4}} \right) \right]\] which is not correct as derivative of \[\left( {{x}^{4}} \right)\] is missing. So that when students use derivatives of a function they have to be more careful. Also, when we write a final answer by rearranging the terms make sure that in the final answer all terms are present or not.