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.