Calculus Examples

$\frac{2 x}{e^{2 x}}$

Step 1

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

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

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Multiply $2$ by $2$ .

$2 \frac{e^{2 x} \frac{d}{d x} [x] - x \frac{d}{d x} [e^{2 x}]}{e^{4 x}}$

Step 3.2

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

$2 \frac{e^{2 x} \cdot 1 - x \frac{d}{d x} [e^{2 x}]}{e^{4 x}}$

Step 3.3

Multiply $e^{2 x}$ by $1$ .

$2 \frac{e^{2 x} - x \frac{d}{d x} [e^{2 x}]}{e^{4 x}}$

Step 4

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 4.1

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

$2 \frac{e^{2 x} - x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])}{e^{4 x}}$

Step 4.2

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

$2 \frac{e^{2 x} - x (e^{u} \frac{d}{d x} [2 x])}{e^{4 x}}$

Step 4.3

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

$2 \frac{e^{2 x} - x (e^{2 x} \frac{d}{d x} [2 x])}{e^{4 x}}$

Step 5

Differentiate.

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Step 5.1

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

$2 \frac{e^{2 x} - x (e^{2 x} (2 \frac{d}{d x} [x]))}{e^{4 x}}$

Step 5.2

Multiply $2$ by $- 1$ .

$2 \frac{e^{2 x} - 2 x (e^{2 x} (\frac{d}{d x} [x]))}{e^{4 x}}$

Step 5.3

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

$2 \frac{e^{2 x} - 2 x (e^{2 x} \cdot 1)}{e^{4 x}}$

Step 5.4

Combine fractions.

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Step 5.4.1

Multiply $e^{2 x}$ by $1$ .

$2 \frac{e^{2 x} - 2 x e^{2 x}}{e^{4 x}}$

Step 5.4.2

Combine $2$ and $\frac{e^{2 x} - 2 x e^{2 x}}{e^{4 x}}$ .

$\frac{2 (e^{2 x} - 2 x e^{2 x})}{e^{4 x}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{2 e^{2 x} + 2 (- 2 x e^{2 x})}{e^{4 x}}$

Step 6.2

Multiply $- 2$ by $2$ .

$\frac{2 e^{2 x} - 4 x e^{2 x}}{e^{4 x}}$

Step 6.3

Reorder terms.

$\frac{- 4 e^{2 x} x + 2 e^{2 x}}{e^{4 x}}$

Step 6.4

Factor $2 e^{2 x}$ out of $- 4 e^{2 x} x + 2 e^{2 x}$ .

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Step 6.4.1

Factor $2 e^{2 x}$ out of $- 4 e^{2 x} x$ .

$\frac{2 e^{2 x} (- 2 x) + 2 e^{2 x}}{e^{4 x}}$

Step 6.4.2

Factor $2 e^{2 x}$ out of $2 e^{2 x}$ .

$\frac{2 e^{2 x} (- 2 x) + 2 e^{2 x} (1)}{e^{4 x}}$

Step 6.4.3

Factor $2 e^{2 x}$ out of $2 e^{2 x} (- 2 x) + 2 e^{2 x} (1)$ .

$\frac{2 e^{2 x} (- 2 x + 1)}{e^{4 x}}$

Step 6.5

Cancel the common factor of $e^{2 x}$ and $e^{4 x}$ .

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Step 6.5.1

Factor $e^{4 x}$ out of $2 e^{2 x} (- 2 x + 1)$ .

$\frac{e^{4 x} (2 e^{- 2 x} (- 2 x + 1))}{e^{4 x}}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Multiply by $1$ .

$\frac{e^{4 x} (2 e^{- 2 x} (- 2 x + 1))}{e^{4 x} \cdot 1}$

Step 6.5.2.2

Cancel the common factor.

$\frac{e^{4 x} (2 e^{- 2 x} (- 2 x + 1))}{e^{4 x} \cdot 1}$

Step 6.5.2.3

Rewrite the expression.

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

Step 6.5.2.4

Divide $2 e^{- 2 x} (- 2 x + 1)$ by $1$ .

$2 e^{- 2 x} (- 2 x + 1)$

Step 6.6

Apply the distributive property.

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

Step 6.7

Rewrite using the commutative property of multiplication.

$2 \cdot - 2 e^{- 2 x} x + 2 e^{- 2 x} \cdot 1$

Step 6.8

Multiply $2$ by $1$ .

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

Step 6.9

Multiply $2$ by $- 2$ .

$- 4 e^{- 2 x} x + 2 e^{- 2 x}$

Step 6.10

Reorder factors in $- 4 e^{- 2 x} x + 2 e^{- 2 x}$ .

$- 4 x e^{- 2 x} + 2 e^{- 2 x}$