Calculus Examples

$2 e^{- x} e^{2 x}$

Step 1

Multiply $e^{- x}$ by $e^{2 x}$ by adding the exponents.

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Step 1.1

Move $e^{2 x}$ .

$\frac{d}{d x} [2 (e^{2 x} e^{- x})]$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [2 e^{2 x - x}]$

Step 1.3

Subtract $x$ from $2 x$ .

$\frac{d}{d x} [2 e^{x}]$

Step 2

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{x}]$ .

$2 \frac{d}{d x} [e^{x}]$

Step 3

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

$2 e^{x}$