Question

How you differentiate $y=\log \left( {{x}^{2}}+1 \right)?$

Answer 1

Hint: Differentiation in mathematics process of finding the derivative or rate of change of a function for solving this question or in other words for differentiating this equation we will use chain rule. The chain rule states that the derivative of composite function is given by
$F(x)=F'\left( g\left( x \right) \right)g'(x)$
$F'\left( g\left( x \right) \right)$ is to differentiate the outer function and $g'(x)$ differentiate the inner function.

Complete step-by-step answer:
We have to differentiate this $y=\log \left( {{x}^{2}}+1 \right)$
As we know that the derivative $\log x=\dfrac{1}{x}$
We will use chain rule here,
$\dfrac{d}{dx}\left[ f\left( g\left( x \right) \right) \right]=f'\left( g\left( x \right) \right)g'(x)$
And the standard derivative of $\dfrac{d}{dx}\left( \log x \right)=\dfrac{1}{x}$
$y=\log \left( {{x}^{2}}+1 \right)$
$\dfrac{dy}{dx}=\dfrac{1}{{{x}^{2}}+1}\dfrac{d}{dx}\left( {{x}^{2}}+1 \right)$
$=\dfrac{1}{{{x}^{2}}+1}.2x$

$\dfrac{dy}{dx}=\dfrac{2x}{{{x}^{2}}+1}$

Additional Information:
Differentiation is one of the two important concepts apart from integration. Differentiation is a method of finding the derivative of a function. Differentiation is a process in maths where we find the instantaneous rate of change in function based on one of its variables. The most common example is the rate of change of displacement with respect to time called velocity. The example can be asked as $y=\ln \left( {{x}^{2}}+1 \right)$ for your better understanding we will take example. i.e. $y=\ln \left( {{x}^{2}}+x \right)$
Here, we will also use chain rules for finding derivatives.
$y=\ln \left( {{x}^{2}}+x \right)$
$\dfrac{dy}{dx}=\dfrac{1}{{{x}^{2}}+x}\left( {{x}^{2}}+x \right)$
$\dfrac{dy}{dx}=\dfrac{1}{{{x}^{2}}+x}\left( 2x+1 \right)$
$\dfrac{dy}{dx}=\dfrac{2x+1}{{{x}^{2}}+1}$
After solving you will get this. In this in place of $\log $ in used. Or we can say that natural log in given in the question.

Note:
While solving this derivative you have learned all the formulas of differentiation. In this numerical we have used chain rule of differentiation is not possible that all questions can be solved by only this rule. So, learn all the rules of differentiation and write the correct formula which you are going to use in the Numerical. Students make mistakes in writing formulas and while solving log problems write carefully.