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Question: The Value of differentiation of \({e^{{x^2}}}\) with respect to \({e^{2x - 1}}\) at \(x = 1\) is ? ...

The Value of differentiation of ex2{e^{{x^2}}} with respect to e2x−1{e^{2x - 1}} at x=1x = 1 is ?
A. ee
B. 00
C. e−1{e^{ - 1}}
D. 11

Explanation

Solution

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)u = f(x) and v=g(x)v = g(x) be two functions of xx . Then to find the derivative of f(x)f(x) with respect to g(x)g(x) that is to find dudv\dfrac{{du}}{{dv}} we will make use of the following formula:
dudv=dudxdvdx\dfrac{{du}}{{dv}} = \dfrac{{\dfrac{{du}}{{dx}}}}{{\dfrac{{dv}}{{dx}}}}

Thus to find the derivative of a function f(x)f(x) with respect to another function g(x)g(x) we differentiate both with respect to xx and then divide the derivative of the function f(x)f(x) with respect to xx by the derivative of the function g(x)g(x) with respect to xx .
Let us define u=f(x)=u = f(x) = ex2{e^{{x^2}}}
Then differentiating both sides with respect to xx we get
dudx=ex22x\dfrac{{du}}{{dx}} = {e^{{x^2}}}2x
Let us define v=g(x)=e2x−1v = g(x) = {e^{2x - 1}}

Then differentiating both the sides with respect to xx we get
dvdx=e2x−1(2)\dfrac{{dv}}{{dx}} = {e^{2x - 1}}(2)
Then we have the following
dudv=dudxdvdx ⇒dudv=ex22xe2x−1(2)\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 xx that is x=1x = 1 we get
dudv=dudxdvdx ⇒dudv=2e2e ∴dudv=1\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.