Question

The derivative of ${e^{{x^3}}}$ with respect to $\log x$ is
1. ${e^{{x^3}}}3{x^3}$
2. $3{x^2}{e^{{x^3}}}$
3. ${e^{{x^3}}}$
4. $3{x^2}{e^{{x^3}}} + 3{x^2}$

Answer 1

An exponential function is defined by the formula $f\left( x \right) = {a^x}$, where the input variable $x$ occurs as an exponent. To find the derivative of ${e^{{x^3}}}$ with respect to $\log x$ we need to consider the given function as $u$ and $v$ and then find the differentiation of $u$ with respect to $v$, and then find the derivative of the given function.

Complete step-by-step solution:
To find the derivative of ${e^{{x^3}}}$ with respect to $\log x$,
Let,
$u = {e^{{x^3}}}$
Then, its derivative is:
$\Rightarrow \dfrac{{du}}{{dx}} = {e^{{x^3}}}3{x^2}$
And, let:
$v = \log x$
Then, its derivative is:
$\Rightarrow \dfrac{{dv}}{{dx}} = \dfrac{1}{x}$
Now, with respect to $u$ and $v$, the derivative is:
$\Rightarrow \dfrac{{du}}{{dv}} = \dfrac{{\left( {\dfrac{{du}}{{dx}}} \right)}}{{\left( {\dfrac{{dv}}{{dx}}} \right)}}$
Substitute the respective values as per obtained i.e.,
$\Rightarrow \dfrac{{du}}{{dv}} = \dfrac{{{e^{{x^3}}}3{x^2}}}{{\dfrac{1}{x}}}$
Hence, differentiating the numerator and denominator terms, we get:
$\Rightarrow \dfrac{{du}}{{dv}} = {e^{{x^3}}}3{x^3}$
Therefore, option $\left( 1 \right)$ is the answer.

Note: The exponential curve depends on the exponential function and it depends on the value of the $x$. The exponential function is an important mathematical function which is of the form $f\left( x \right) = {a^x}$. The base of the exponential function is encountered by , which is approximately equal to $2.71828$.