Calculus Examples

$y = \frac{1}{e^{2 x}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Rewrite $\frac{1}{e^{2 x}}$ as ${(e^{2 x})}^{- 1}$ .

$\frac{d}{d x} [{(e^{2 x})}^{- 1}]$

Step 1.2

Multiply the exponents in ${(e^{2 x})}^{- 1}$ .

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Step 1.2.1

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

$\frac{d}{d x} [e^{2 x \cdot - 1}]$

Step 1.2.2

Multiply $- 1$ by $2$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $- 2 x$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{- 2 x} (- 2 \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 2$ by $1$ .

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

Step 3.3.2

Move $- 2$ to the left of $e^{- 2 x}$ .

$- 2 e^{- 2 x}$