Question

What is the derivative of ${{e}^{1}}$?

Answer 1

For solving this question you should know about the differentiation of exponential functions and how to calculate the derivatives. In this question we will differentiate to ${{e}^{1}}$ with respect to any variable. But as we know that ${{e}^{1}}$ is a constant value and the differentiation of the constant is 0 because it will never change, so the rate of change of ${{e}^{1}}$ function will be always zero.

Complete step by step answer:
According to our question , it is asked of us to determine the derivative of ${{e}^{1}}$. So, as we know that the differentiation of any exponential function will be as $\dfrac{d}{dx}{{e}^{x}}={{e}^{x}}$, so we can say that the derivative of are always same as exponential term $\left( {{e}^{x}} \right)$. If we see examples of this, then:
Example: Find the derivative of ${{e}^{2x}}$.
We have to find the derivative of ${{e}^{2x}}$. So, as we know that $\dfrac{d}{dx}{{e}^{ax}}=a.{{e}^{ax}}$, so,
$\dfrac{d}{dx}{{e}^{2x}}=2.{{e}^{2x}}$
But if we see our question, then we know that ${{e}^{1}}=2.7182818284$ which is a constant value. And we know that the differentiation of the constant is always zero. And the derivatives of the constants are always zero because they do not change with the variable in whose respect they are going to be differentiated. So, the differentiation of ${{e}^{1}}$,
$\begin{aligned} & \Rightarrow \dfrac{d}{dx}{{e}^{1}}=? \\\ & \Rightarrow \dfrac{d}{dx}\left( 2.71 \right)=0 \\\ \end{aligned}$

So, the derivative of ${{e}^{1}}$ is equal to zero.

Note: During solving the differentiation of any term we always have to be assure that the term which we are differentiating and the variable with whose respect we differentiate to this, always have the same variable, unless this will be a constant for that and the differentiation will be zero.