Question

How do you differentiate $y = 2{e^x}?$

Answer 1

Hint : As we know that to differentiate means to find the derivative of independent variable value which changes the value of the function. Let a function be $y = f(x)$ , where $y$ is a function of $x$ . Any change in the value of $y$ due to the change in the value of $x$ will be written as $\dfrac{{dy}}{{dx}}$ . This is the general expression of the derivative of a function.

Complete step-by-step answer :
The derivative of the second part ${e^x}$ is itself only, it is Euler’s number and we know that $\dfrac{d}{{dx}}({e^x}) = {e^x}$ and the constant part just comes out of the derivative.
So we have
$\dfrac{d}{{dx}}[2{e^x}] = 2\dfrac{d}{{dx}}[{e^x}]$ , as the constant part comes out which gives us $\dfrac{{dy}}{{dx}} = 2{e^x}$ .
We have nothing to do with chain rule here because if we further change it in the term $u$ and $\dfrac{d}{{dx}}({e^u}) = {e^u}(\dfrac{{du}}{{dx}})$ but we know that $u(x) = x$ and $\dfrac{{du}}{{dx}} = 1$ , so at last we get
$\dfrac{d}{{dx}}({e^x}) = {e^x}*1 = {e^x}$ .
It gives the same result. We do not need to apply chain rule here.
Hence the differentiation of $y = 2{e^x}$ is $2{e^x}$ .
So, the correct answer is “ $2{e^x}$ ”.

Note : We should know that ${e^x}$ is an exponential function and the base of this function is $e,$ Euler’s number which is the only function that remains unchanged when differentiated. It is an irrational number and is approximately $2.7182$ . In the above question $2$ is a constant and any constant which is multiplied by a variable remains the same when taking a derivative. We should first understand what is asked in the question and then proceed to differentiate the function in the right way to get the correct answer.