Question

Derivative of ${\log }_{10}\left(x\right)$ with respect to $x^2$is

• A. $2x^2{\log }_e\left(10\right)$
• B. $\frac{{\log }_{10}\left(e\right)}{2x^2}$
• C. $\frac{{\log }_e\left(10\right)}{2x^2}$
• D. $x^2{\log }_e\left(10\right)$

Answer 1

The correct option is B

$\frac{{\log }_{10}\left(e\right)}{2x^2}$

Explanation for the correct option:

Step-1: Differentiate ${\log }_{10}\left(x\right)$ with respect to $x$.

Let, $u={\log }_{10}\left(x\right)$

$\Rightarrow u=\frac{{\log }_e\left(x\right)}{{\log }_e\left(10\right)}$

Differentiate w.r.t. $x$

$$\begin{gathered} \Rightarrow \frac{du}{dx}=\frac{1}{{\log }_e\left(10\right)}\times \frac{1}{x} \\ \Rightarrow \frac{du}{dx}=\frac{1}{x{\log }_e\left(10\right)}............\left(i\right) \end{gathered}$$

Step-2: Differentiate $x^2$with respect to $x$.

Let, $v=x^2$

Differentiate w.r.t. $x$

$\Rightarrow \frac{dv}{dx}=2x...........\left(ii\right)$

Step-3: Divide the equation $\left(i\right)$ by $\left(ii\right)$.

$$\begin{gathered} \Rightarrow \frac{{\displaystyle \frac{du}{dx}}}{{\displaystyle \frac{dv}{dx}}}=\frac{{\displaystyle \frac{1}{x{\log }_e\left(10\right)}}}{2x} \\ \Rightarrow \frac{du}{dv}=\frac{1}{2x^2{\log }_e\left(10\right)} \\ \Rightarrow \frac{du}{dv}=\frac{{\log }_{10}\left(e\right)}{2x^2} \end{gathered}$$

Hence, the correct answer is option $\left(B\right)$