Question

What is the derivative of $\sec \left( {{x}^{2}} \right)$ ?

Answer 1

Hint: To solve this question we need to know the concept of differentiation and its formula. The method used here to differentiate the given question is usage of chain rule which has formula as ${{y}^{'}}={{f}^{'}}\left( g\left( x \right) \right).{{g}^{'}}\left( x \right)$ , where $f\left( x \right)$ and $g\left( x \right)$ are the two functions.

Complete step by step solution:
The question asks us to find the derivation of the function $\sec \left( {{x}^{2}} \right)$. The angle substituted by the trigonometric function sec is ${{x}^{2}}$ . The method used here to solve the problem is using the chain rule of differentiation. The chain rule method helps us differentiate composite functions. As that is given in the question. So here $\sec \left( {{x}^{2}} \right)$ is a composite function because it can be constructed as $f\left( u \right)$ for $f\left( u \right)=\sec \left( u \right)$ and $u={{x}^{2}}$. On applying the formula \[y'=f'(u)\cdot u'(x)\] to the function given.
The first term to be differentiated in the function is $\sec {{x}^{2}}$. On differentiating the function $\sec \left( u \right)$ with respect to $u$ the formula used is$\dfrac{d\left( \sec u \right)}{du}=\sec u.\tan u$. Applying the same in the formula where $u={{x}^{2}}$, we get:
$\Rightarrow \dfrac{d\left( \sec {{x}^{2}} \right)}{dx}$
$\Rightarrow \dfrac{d\left( \sec {{x}^{2}} \right)}{dx}=\left( \sec {{x}^{2}} \right)\left( \tan {{x}^{2}} \right)$
As we have considered $f\left( x \right)=\sec \left( u \right)$ and $g\left( x \right)={{x}^{2}}$so the second term that need to be differentiated is ${{x}^{2}}$. On differentiating the function $u$ which is ${{x}^{2}}$ with respect to $x$, we get:
$\Rightarrow \dfrac{d\left( {{x}^{2}} \right)}{dx}$
The formula used to evaluate the above integral is $\dfrac{d\left( {{x}^{n}} \right)}{dx}=n{{x}^{n-1}}$ . In the above expression the value of “n” is $2$ , so on applying the formula we get:
$\Rightarrow 2x$
On combining the two differentiation we get:
$\Rightarrow \left( \sec {{x}^{2}} \right)\left( \tan {{x}^{2}} \right)2x$

$\therefore $ The derivative of $\sec {{x}^{2}}$ is $\left( \sec {{x}^{2}} \right)\left( \tan {{x}^{2}} \right)2x$.

Note: To solve the question on differentiation we need to know the different type of methods of differentiation, and the places of their application. The formula of the differentiation should be known so the calculation becomes a bit easier.