Question

The Value of differentiation of ${e^{{x^2}}}$ with respect to ${e^{2x - 1}}$ at $x = 1$ is ?
A. $e$
B. $0$
C. ${e^{ - 1}}$
D. $1$

Answer 1

Hint: in the given question we are asked to find the derivative of a function with respect to another function. Do not misinterpret this type of question with the one in which we are asked to find the derivative of a function with respect to the given variable. find the respective derivatives of the given functions with respect to the given variable separately.

Complete step by step answer:
Here we will use the concept of finding the derivative of a function with respect to another function. Let $u = f(x)$ and $v = g(x)$ be two functions of $x$ . Then to find the derivative of $f(x)$ with respect to $g(x)$ that is to find $\dfrac{{du}}{{dv}}$ we will make use of the following formula:
$\dfrac{{du}}{{dv}} = \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}}$

Thus to find the derivative of a function $f(x)$ with respect to another function $g(x)$ we differentiate both with respect to $x$ and then divide the derivative of the function $f(x)$ with respect to $x$ by the derivative of the function $g(x)$ with respect to $x$ .
Let us define $u = f(x) = $${e^{{x^2}}}$
Then differentiating both sides with respect to $x$ we get
$\dfrac{{du}}{{dx}} = {e^{{x^2}}}2x$
Let us define $v = g(x) = {e^{2x - 1}}$

Then differentiating both the sides with respect to $x$ we get
$\dfrac{{dv}}{{dx}} = {e^{2x - 1}}(2)$
Then we have the following
$\dfrac{{du}}{{dv}} = \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}} \\
\Rightarrow \dfrac{{du}}{{dv}}= \dfrac{{{e^{{x^2}}}2x}}{{{e^{2x - 1}}(2)}}$
Putting the given value of $x$ that is $x = 1$ we get
$\dfrac{{du}}{{dv}} = \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}} \\
\Rightarrow \dfrac{{du}}{{dv}}= \dfrac{{2e}}{{2e}} \\
\therefore \dfrac{{du}}{{dv}}= 1$

Therefore option A is the correct answer.

Note: Find the respective derivatives of the given functions with respect to the given variable separately.The derivative function gives the derivative of a function at each point in the domain of the original function for which the derivative is defined. Do not forget to find the particular solution at the end of the general solution.