Question

What is the derivative of \[{{e}^{-1}}\]?

Answer 1

Hint: 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 \[{{e}^{-1}}\] with respect to any variable. But as we know that the \[{{e}^{-1}}\] is a constant value and differentiation of 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, if we see that it is asked to determine the derivative of \[{{e}^{-1}}\] or \[\dfrac{1}{e}\].
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 derivatives of \[{{e}^{x}}\] are always the same as the exponential term (\[{{e}^{x}}\]). If we see examples of this, then-
Example (1) Find the derivative of \[{{e}^{2x}}\].
Solution- We have to find the derivative of \[{{e}^{2x}}\].
So, as we know that the \[\dfrac{d}{dx}{{e}^{ax}}=a.{{e}^{ax}}\].
So, \[\dfrac{d}{dx}{{e}^{2x}}=2{{e}^{2x}}\]
But if we see to our question then, we know that the \[{{e}^{1}}=2.7182818284\] which is a constant value and \[{{e}^{-1}}\] is equal to \[\dfrac{1}{e}\], and the value of this is equal to \[\dfrac{1}{2.7182}\] which will be a constant.
And we know that the differentiation of the constant is always zero.
And the derivatives of constants are always zero because they do not change with the variable in whose respect they are going to differentiate.
So, the differentiation of \[{{e}^{-1}}\]:
\[\begin{align}
  & \Rightarrow \dfrac{d}{dx}{{e}^{-1}}=\dfrac{d}{dx}\left( \dfrac{1}{e} \right)=? \\
 & \Rightarrow \dfrac{d}{dx}\left( \dfrac{1}{2.71} \right)=0 \\
\end{align}\]
So, the derivative of \[{{e}^{-1}}\] is equal to zero.

Note: During solving the differentiation of any term we always have to 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.