Question

How do you find the derivative of $y=\log \left( {{x}^{2}}+1 \right)$?

Answer 1

In this problem, we have to find the derivative of the given expression. To solve this problem, we have to know to differentiate the given expression, we should know the differentiation formulas. We can first differentiate the logarithmic function and then we can differentiate the terms inside the bracket to get the derivative of the given expression. We can use chain rule to find the derivative.

Complete step by step answer:
We know that the given expression to be differentiated is,
$y=\log \left( {{x}^{2}}+1 \right)$…… (1)
In the above expression, we can differentiate the logarithmic function first and then we can differentiate the terms inside the brackets.
Differentiating logarithmic function, we get, the standard derivative is $\dfrac{d}{dx}\log x=\dfrac{1}{x}$ .
Differentiating the terms square of x, we get the standard derivative is ${{x}^{2}}\dfrac{d}{dx}=2x$
We know that the chain rule is,
$\dfrac{d}{dx}\left[ f\left( g\left( x \right) \right) \right]=f'\left( g\left( x \right) \right).g'\left( x \right)$
Now we can differentiate the above given expression (1) using the above formulas by applying the chain rule method, we get
$\Rightarrow y=\log \left( {{x}^{2}}+1 \right)$
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{{{x}^{2}}+1}2x$ .
Therefore, the derivative of the given expression $y=\log \left( {{x}^{2}}+1 \right)$ is $\dfrac{1}{{{x}^{2}}+1}2x$.

Note:
Students make mistakes while differentiating the given expression, we should know some basic differentiation formulas to solve or find the derivative for these types of problems. In this problem, we have used some differentiating formula such as, ${{x}^{2}}\dfrac{d}{dx}=2x$, $\dfrac{d}{dx}\log x=\dfrac{1}{x}$ which we can use in these types of problems. We can also use chain rule to find the derivative and solve the given question.