Question

How do you take the derivative of ${{\tan }^{-1}}\left( {{x}^{2}} \right)$ ?

Answer 1

Derivative is the rate of change of a function. To find the derivative of a function, we have to differentiate it with respect to $x$ . Finding out the derivative of a function implies that we are finding out the slope of the function. Here we have to use the Extension Chain Rule to solve this. We use this rule when the function , let’s say $y$, is a function of $u$ and $u$ is a function of $v$ and $v$ is a function of $x$. It states the following : $\dfrac{dy}{dx}=\dfrac{dy}{du}\times \dfrac{du}{dv}\times \dfrac{dv}{dx}$ . So when we differentiate a function like that, we have to use this formula.

Complete step by step answer:
Here $y={{\tan }^{-1}}\left( {{x}^{2}} \right)$ , it is a function of ${{\tan }^{-1}}$ and ${{\tan }^{-1}}$ is a function of ${{x}^{2}}$.So we have to apply the Extension Chain rule in order to find the derivative of the given function. Upon comparing , $u$ is ${{\tan }^{-1}}x$ and $v$ is ${{x}^{2}}$ .
We already know that the derivative of ${{\tan }^{-1}}x$ is
$\Rightarrow$ $\dfrac{d\left( {{\tan }^{-1}}x \right)}{dx}=\dfrac{1}{1+{{x}^{2}}}$ ( This is the derivative of ${{\tan }^{-1}}x$.)
$\Rightarrow \dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ which is the derivative of ${{x}^{n}}$
Here $n=2$ , so $\dfrac{d\left( {{x}^{2}} \right)}{dx}=2x$ ……. $eqn\left( 1 \right)$
Upon differentiating $y$with respect to $x$, we get the following :
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( {{\tan }^{-1}}{{x}^{2}} \right)}{dx}\times \dfrac{d{{\left( {{x}^{2}} \right)}^{{}}}}{dx}$
From $eqn\left( 1 \right)$ :
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{4}}}\times 2x$.
And we can further write it as
$\Rightarrow \dfrac{dy}{dx}=\dfrac{2x}{1+{{x}^{4}}}$ .
$\dfrac{dy}{dx}$ here is nothing but the slope . So in other words the slope of the function ${{\tan }^{-1}}\left( {{x}^{2}} \right)=\dfrac{2x}{1+{{x}^{4}}}$

$\therefore$ Hence , the derivative of ${{\tan }^{-1}}\left( {{x}^{2}} \right)$ = $\dfrac{2x}{1+{{x}^{4}}}$.

Note: In this question we have to be careful about a lot of things. First we should remember the derivatives of all function such as $\sin x,\cos ,\tan x,{{x}^{n}},\log x,{{a}^{x}},{{\sin }^{-1}}x,{{\cos }^{-1}}x,{{\tan }^{-1}}x$ and all other related functions so that we can solve it easily. Secondly we should be careful while applying the Extension Chain Rule. If we get confused in applying, then the whole derivative may be wrong. It is also advisable to remember all the other standard rules of differentiation.