Question

What is the derivative of $y=\arctan \left( x \right)$?

Answer 1

In this question we have been given the term $\arctan \left( x \right)$ which we have to differentiate. We will consider the given term to be $y$ and then we will use implicit differentiation on both the sides of the expression. After using implicit derivation, we will rearrange the term in terms of $\dfrac{dy}{dx}$ and then substitute the actual value of $y$ to get the required solution.

Complete step by step solution:
We have the term given to us as:
$y=\arctan \left( x \right)$
Now we know that $\arctan \left( x \right)$ is the inverse of $\tan \left( x \right)$ therefore, on taking the tan on both the sides, we get:
$\Rightarrow \tan \left( y \right)=\tan \left( \arctan \left( x \right) \right)$
Now we know the trigonometric property that $\tan \left( \arctan \left( x \right) \right)=x$ therefore, on substituting, we get:
$\Rightarrow \tan \left( y \right)=x\to \left( 1 \right)$
Now we will use implicit differentiation with respect to $x$ on both the sides of the expression therefore, we get:
$\Rightarrow \dfrac{d}{dx}\tan \left( y \right)=\dfrac{dx}{dx}$
Now we know that $\dfrac{d}{dx}\tan \left( x \right)={{\sec }^{2}}\left( x \right)$ and $\dfrac{dx}{dx}=1$ therefore, we get:
$\Rightarrow {{\sec }^{2}}\left( y \right)\dfrac{dy}{dx}=1$
Now on transferring the term ${{\sec }^{2}}\left( y \right)$ from the right-hand side to the left-hand side, we get:
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{{{\sec }^{2}}\left( y \right)}$
Now we know the trigonometric property that ${{\sec }^{2}}\left( x \right)=1+{{\tan }^{2}}\left( x \right)$ therefore, on substituting, we get:
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{1+{{\tan }^{2}}\left( y \right)}$
Now from equation $\left( 1 \right)$, we know that $\tan \left( y \right)=x$ therefore, on substituting, we get:
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{1+{{x}^{2}}}$, which is the required derivative.
Therefore, we can write:
$\Rightarrow \dfrac{d}{dx}\arctan \left( x \right)=\dfrac{1}{1+{{x}^{2}}}$, which is the required solution.

Note: It is to be remembered that this derivative should be remembered as a direct formula for the derivative of the inverse function of $\arctan \left( x \right)$. In the question we used the property $\tan \left( \arctan \left( x \right) \right)=x$, the formula $\arctan \left( \tan x \right)=x$ should also be remembered. It is to be noted that in some questions $\arctan \left( x \right)$ can be written as ${{\tan }^{-1}}\left( x \right)$.