Question

What is the derivative of ${{e}^{-kx}}$?

Answer 1

We are given a question based on differentiation. We are given an exponential function which we have to differentiate with respect to x. We know that the differentiation of an exponential function is exponential itself, that is, $\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$. So, we will first differentiate the given function, then we will differentiate the powers of the exponential function. In other words, we will be using the chain rule to compute the differentiation of the given function. Hence, we will have the value of the derivative of the given function.

Complete step-by-step solution:
According to the given question, we are given an exponential function, which we have to differentiate and find the value. We will proceed to find the derivative of the given exponential function using the chain rule.
We will first let the given function be named (say ‘y’), we get,
$y={{e}^{-kx}}$-----(1)
We will now differentiate both the sides of the equation (1) and we get,
$\dfrac{dy}{dx}=\dfrac{d}{dx}\left( {{e}^{-kx}} \right)$------(2)
We now have the new expression as,
$\Rightarrow \dfrac{dy}{dx}={{e}^{-kx}}\dfrac{d}{dx}\left( -kx \right)$-----(3)
We got the above expression as we know that the differentiation of an exponential function is exponential function itself, that is, $\dfrac{d}{dx}\left( {{e}^{x}} \right)={{e}^{x}}$. So, we had first differentiated the exponential function first, next we will differentiate the powers which the exponential function is raised to.
We get,
$\Rightarrow \dfrac{dy}{dx}=-k{{e}^{-kx}}$------(4)
Therefore, the value of the derivative of the given exponential function is $\dfrac{d}{dx}\left( {{e}^{-kx}} \right)=-k{{e}^{-kx}}$.

Note: The differentiation is always to be computed in a stepwise manner. Also, most likely the mistake occurs when students forget to differentiate the power which the exponential function is raised to. When the power is still left to differentiate, we only have with us half the correct answer or in other words, a wrong answer. Care must be taken while using the chain rule.