Question

The derivative of $e^{x^3}$ with respect to$\log x$ is

• A. $e^{x^3}3x^3$
• B. $3x^2{e}^{x^3}$
• C. $e^{x^3}$
• D. $3x^2{e}^{z3}+3x^2$

Answer 1

The correct option is A

$e^{x^3}3x^3$

Explanation for correct option

Let $u=e^{x^3}$ and $v=\log x$

Differentiating $u=e^{x^3}$ With respect to $x$ we get,

$$\begin{gathered} \frac{d}{dx}\left(u\right)=\frac{du}{dx} \\ =\frac{d}{dx}\left(e^{x^3}\right) \\ =e^{x^3}\times \frac{d}{dx}\left(x^3\right) \\ =e^{x^3}\times 3x^2\left(ApplyingChainRule\right) \\ =3x^2{e}^{x^3}......\left(1\right) \end{gathered}$$

Differentiating $v=\log x$ With respect to $x$ we get,

$$\begin{gathered} \frac{d}{dx}\left(v\right)=\frac{dv}{dx} \\ =\frac{d}{dx}\left(x\right) \\ =\frac{1}{x}......\left(2\right) \end{gathered}$$

Now dividing the Equation$\left(1\right)$ and Equation $\left(2\right)$ we have,

$$\begin{gathered} \frac{du}{dv}=\left(\frac{{\displaystyle \frac{du}{dx}}}{{\displaystyle \frac{dv}{dx}}}\right) \\ =\frac{3x^2{e}^{x^3}}{\left({\displaystyle \frac{1}{x}}\right)} \\ =\left(x\right)\left(3x^2{e}^{x^3}\right) \\ =3x^3{e}^{x^3} \end{gathered}$$

Hence, the correct answer is Option (A).