Question

How do you differentiate $y = 5{e^x}$ ?

Answer 1

Hint : We are given a function in terms of x, the function is an exponential function. Here, x is the independent variable and y is the dependent variable, that is, the value of y depends on the value of the x. Now, we have to find the derivative of this function, so we must know the definition of differentiation. The process of dividing a whole quantity into very small ones is known as differentiation, in the given question we have to differentiate y with respect to x.

Complete step-by-step answer :
We are given that $y = 5{e^x}$ and we have to differentiate the function y.
We know that the differentiation of the product of a constant and a function is equal to the product of the constant and the derivative of the function. So,
$\dfrac{{dy}}{{dx}} = \dfrac{{d(5{e^x})}}{{dx}} \\\ \Rightarrow \dfrac{{dy}}{{dx}} = 5\dfrac{{d{e^x}}}{{dx}} \;$
Now, ${e^x}$ is a transcendental function, that is, its value remains unchanged after differentiation. So,
$\dfrac{{dy}}{{dx}} = 5{e^x}$
Hence the derivative of the given function remains unchanged, that is, $5{e^x}$
So, the correct answer is “ $5{e^x}$ ”.

Note : Usually, the rate of change of something is observed over a specific duration of time, but if we have to find the instantaneous rate of change of a quantity then we differentiate it, in the expression $\dfrac{{dy}}{{dx}}$ , $dy$ represents a very small change in the quantity and $dx$ represents the small change in the quantity with respect to which the given quantity is changing. In the given question, we have a function of x, so by putting different values of x, we can find the instantaneous change in x at that particular value.