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.