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. \[3{x^2}{e^{{x^3}}} + 3{x^2}\]

Hint: 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 \[2.71828\].