What is the Derivative of 1/e^x?

derivative of inverse exponential

What is the derivative of 1/e^x?

The derivative of

\frac{1}{e^x}

-\frac{1}{e^x}

.

Solution. Let

h(x) = \frac{1}{e^x} = e^{-x}

f(u) = e^u

and

g(x) = -x

. We will use the chain rule, i.e.,

\begin{align*} h'(x) = f'(g(x))g'(x). \end{align*}

We know that

\frac{d}{dx} e^x = e^x

and

\frac{d}{dx} -x = -1

. So we get

\begin{align*} f(u) = e^u \quad \text{and} \quad g(x) = -1. \end{align*}

Combining everything, we get

\begin{align*} h'(x) &= f'(g(x))g'(x) \\ &= e^{-x} \cdot (-1) \\ &= -e^{-x} \\ &= -\frac{1}{e^{-x}} \end{align*}

So the derivative of

\frac{1}{e^x}

-\frac{1}{e^x}