Question

What is the derivative of $c.\left( {{e}^{x}} \right)$ where c is a constant?

Answer 1

Hint: Assume the given function as y. Now, consider y as the product of two functions and apply the product rule of differentiation given as: - \[\dfrac{d\left( u\times v \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\]. Here, consider \[u=c\] and \[v={{e}^{x}}\]. Use the formula \[\dfrac{d{{e}^{x}}}{dx}={{e}^{x}}\] and the fact that the derivative of a constant is 0 to get the answer.

Complete step by step solution:
Here we have been provided with the function $c.\left( {{e}^{x}} \right)$, c being a constant, and we are asked to find its derivative. Here we are going to use the product rule of derivative to get the answer. Let us assume the given function as y. So we have,
\[\Rightarrow y=c.\left( {{e}^{x}} \right)\]
Now, we can consider the above expression as the product of a constant function (c) and an exponential function (\[{{e}^{x}}\]). Let us assume constant c as u and ${{e}^{x}}$ as v respectively. So we have,
\[\Rightarrow y=u\times v\]
Differentiating both sides with respect to x, we get,
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left( u\times v \right)}{dx}\]
Now, applying the product rule of differentiation given as \[\dfrac{d\left( u\times v \right)}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\], we get,
\[\Rightarrow \dfrac{dy}{dx}=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\]
Substituting the assumed values of u and v, we get,
\[\Rightarrow \dfrac{dy}{dx}=c\dfrac{d\left( {{e}^{x}} \right)}{dx}+{{e}^{x}}\dfrac{d\left( c \right)}{dx}\]
We know that \[\dfrac{d{{e}^{x}}}{dx}={{e}^{x}}\] and the derivative of a constant function is 0, so we get,
\[\begin{align}
  & \Rightarrow \dfrac{dy}{dx}=c\left( {{e}^{x}} \right)+{{e}^{x}}\left( 0 \right) \\
 & \Rightarrow \dfrac{dy}{dx}=c{{e}^{x}}+0 \\
 & \therefore \dfrac{dy}{dx}=c{{e}^{x}} \\
\end{align}\]
Hence, the above relation is our answer.

Note: From the above question you may note an important formula given as $\dfrac{d\left[ kf\left( x \right) \right]}{dx}=k\dfrac{d\left[ f\left( x \right) \right]}{dx}$. It can be explained as: - if a constant is multiplied to a function then that constant can be taken out from the derivative formula. You must remember all the basic rules and formulas of differentiation like: - product rule, chain rule, \[\dfrac{u}{v}\] rule etc. Remember the derivative formulas of the functions like exponential, logarithmic, trigonometric functions etc.