Question

What is the derivative of ${{e}^{a}}$ ( a is any constant number)?

Answer 1

First we will assume that y is the given function i.e. $y={{e}^{a}}$ and we have to find the value of $\dfrac{dy}{dx}$. We will apply the differentiation rule to the given function and find its derivative. We will also consider this as a constant and solve the question accordingly.

Complete step by step solution:
We have been given a function ${{e}^{a}}$. We also have the information that a is a constant. We have been asked to find the derivative of the given function.
Now, let us assume that the given function is $y={{e}^{a}}$.
Now, as given in the question a is a constant number and we know that e is also a constant. The value of e roughly comes up to 2.718.
So we know that differentiation of a constant is always zero. The function ${{e}^{a}}$ is also a constant.
So by applying the differentiation rule $\dfrac{d}{dx}k=0$, where k is a constant we will get
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}{{e}^{a}} \\\ & \Rightarrow \dfrac{dy}{dx}=0 \\\ \end{aligned}$
Hence the derivative of ${{e}^{a}}$ is zero.

Note: If the given function is like ${{e}^{x}}$ then the differentiation of the function is different. Here ${{e}^{x}}$ is the function of x and x is not a constant. Then the derivative of the function will be
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=\dfrac{d}{dx}{{e}^{x}} \\\ & \Rightarrow \dfrac{dy}{dx}={{e}^{x}} \\\ \end{aligned}$
Differentiation defined as the instantaneous rate of change of a function with respect to one of the variables. The value of constant remains the same so the differentiation of constant is always zero.