Question

Find the value of differentiation of ${{e}^{{{x}^{3}}}}$ with respect to log x
(a) ${{e}^{{{x}^{3}}}}$
(b) $3{{x}^{2}}{{e}^{{{x}^{3}}}}+3{{x}^{2}}$
(c) $3{{x}^{3}}{{e}^{{{x}^{3}}}}$
(d) $3{{x}^{2}}{{e}^{{{x}^{3}}}}$

Answer 1

Hint: Here we may use the concept that the derivative of ${{e}^{{{x}^{3}}}}$ with respect to log x is given as derivative of ${{e}^{{{x}^{3}}}}$ with respect to x divided by the derivative of log x with respect to x and we may use chain rule to find the derivative of ${{e}^{{{x}^{3}}}}$ with respect to x.

Complete step-by-step answer:
Since, we have to find the derivative of ${{e}^{{{x}^{3}}}}$ with respect to log x. So, we have:
$\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{d\left( \log x \right)}$
We can also write it as:
\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}\times \dfrac{dx}{d\left( \log x \right)}=\dfrac{\left\\{ \dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx} \right\\}}{\left\\{ \dfrac{d\left( \log x \right)}{dx} \right\\}}
It means that first we may find the derivative of ${{e}^{{{x}^{3}}}}$ with respect to x and in the next step, we can find the derivative of log x with respect to x and then divide the derivative of ${{e}^{{{x}^{3}}}}$ by the derivative of log x.
So, let us first find $\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}$.
Let us take ${{x}^{3}}=y$.
On differentiating it both sides with respect to x, we get:
$3{{x}^{2}}=\dfrac{dy}{dx}........(1)$
So, now using this value we have:
$\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}=\dfrac{d\left( {{e}^{y}} \right)}{dy}\times \dfrac{dy}{dx}={{e}^{y}}\dfrac{dy}{dx}........(2)$
On substituting the value from equation (1) in equation (2) we have:
$\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}={{e}^{y}}\times 3{{x}^{2}}.......(3)$
On substituting $y={{x}^{3}}$ in equation (3) , we get:
$\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}={{e}^{{{x}^{3}}}}\times 3{{x}^{2}}...........(4)$
Now, we will find the derivative of log x with respect to x. So:
$\dfrac{d\left( \log x \right)}{dx}=\dfrac{1}{x}.........(5)$
Since, we already have:
$\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{d\left( \log x \right)}=\dfrac{\dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{dx}}{\dfrac{d\left( \log x \right)}{dx}}............(6)$
On substituting the respective values from equation (4) and (5) in equation (6) , we get:
$\begin{aligned} & \dfrac{d\left( {{e}^{{{x}^{3}}}} \right)}{d\left( \log x \right)}=\dfrac{{{e}^{{{x}^{3}}}}\times 3{{x}^{2}}}{\dfrac{1}{x}} \\\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{e}^{{{x}^{3}}}}\times 3{{x}^{2}}\times x \\\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=3{{x}^{3}}.{{e}^{{{x}^{3}}}} \\\ \end{aligned}$
Hence, option (c) is the correct answer.

Note: Students should note here that chain rule can be applied for finding the derivative of any composite function (a function having another function substituted into the function). So, in this case ${{e}^{{{x}^{3}}}}$ is the composite function of the form ${{e}^{y}}$ where $y={{x}^{3}}$. ${{e}^{{{x}^{3}}}}$