What is the derivative of e^5?

Aug 23, 2015

The derivative is $0$

Explanation:

Here are three ways to see that the derivative is $0$:

The power rule and chain rule

$\frac{d}{\mathrm{dx}} \left({u}^{5}\right) = 5 {u}^{4} \frac{d}{\mathrm{dx}} \left(u\right)$

In this case $u = e$ is a constant, so we get:

$\frac{d}{\mathrm{dx}} \left({e}^{5}\right) = 5 {e}^{4} \frac{d}{\mathrm{dx}} \left(e\right) = 5 {e}^{4} \cdot 0 = 0$

Exponential function and chain rule

$\frac{d}{\mathrm{dx}} \left({e}^{u}\right) = {e}^{u} \frac{d}{\mathrm{dx}} \left(u\right)$

In this case $u = 5$ is a constant, so we get:

$\frac{d}{\mathrm{dx}} \left({e}^{5}\right) = {e}^{5} \frac{d}{\mathrm{dx}} \left(5\right) = {e}^{5} \cdot 0 = 0$

${e}^{5}$ is a constant

$e \approx 2.7$, so ${e}^{5}$ is s a number close to ${2.7}^{5}$.

The derivative of that number (a constant) is $0$

$\frac{d}{\mathrm{dx}} \left({e}^{5}\right) = 0$

Additional note This is a lot like asking for the derivative of ${2}^{5}$ which is clearly the same as the derivative of $32$ which is $0$.

The constant $e$ causes confusion until a student gets comfortable with the fact that $e$ is just some number.

Asking about the derivative of ${x}^{e}$ also causes confusion .