Calculus Examples

$\frac{1}{e^{- x}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{e^{- x}}$ as ${(e^{- x})}^{- 1}$ .

$\frac{d}{d x} [{(e^{- x})}^{- 1}]$

Step 1.2

Multiply the exponents in ${(e^{- x})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [e^{- x \cdot - 1}]$

Step 1.2.2

Multiply $- x \cdot - 1$ .

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Step 1.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [e^{1 x}]$

Step 1.2.2.2

Multiply $x$ by $1$ .

$\frac{d}{d x} [e^{x}]$

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x}$