Question

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

Answer 1

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.