Question

How do you find the derivative of $f(x)=5{{e}^{x}}$?

Answer 1

We calculate the derivative of any function using the method of differentiation . A derivative of a function $f(x)$ with respect to $x$ is the change in the value of a function with the respective change in the value of $x$. We can note that the dx represents an infinitesimal change in the value of $x$ and the derivative of the function is denoted by $\dfrac{df}{dx}$ which can also be represented as $f'(x)$ indicating the derivative of $f(x)$ with respect to $x$. It should also be noted that the derivative of any constant is zero. We generally use some derivative rules to find the derivative of a function. According to the rule, the derivative of an exponential function gives the same value for example the derivative of ${{e}^{x}}$ is ${{e}^{x}}$.

Complete step by step solution:
We have to find the derivative of the function $f(x)=5{{e}^{x}}$.
We will do this by differentiation.
So, $\dfrac{df}{dx}=\dfrac{d5{{e}^{x}}}{dx}$
We can take $5$ out of the derivative symbol because $5$ is present as multiplicative constant coefficient of ${{e}^{x}}$ .
$\dfrac{df}{dx}=5\dfrac{d{{e}^{x}}}{dx}$
Also by the derivative rules , we know that the derivative of ${{e}^{x}}$ is ${{e}^{x}}$
So after differentiation , we get
$\dfrac{df}{dx}=5{{e}^{x}}$

The derivative of $5{{e}^{x}}$ is $5{{e}^{x}}$.

Note:
We can also prove the result of the derivative of an exponential function using a logarithmic function. We can find second, third, fourth, and so on derivatives of the function until we get the constant as the derivative but we will never get the derivative of ${{e}^{x}}$ as constant as it continues to return the same value.