Calculus Examples

$\frac{e^{- x}}{- 3 x}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{e^{- x}}{3 x}]$

Step 1.2

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{e^{- x}}{3 x}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [\frac{e^{- x}}{x}]$ .

$- \frac{1}{3} \frac{d}{d x} [\frac{e^{- x}}{x}]$

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 $f (x) = e^{- x}$ and $g (x) = x$ .

$- \frac{1}{3} \cdot \frac{x \frac{d}{d x} [e^{- x}] - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 3

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 $g (x) = - x$ .

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Step 3.1

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

$- \frac{1}{3} \cdot \frac{x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

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

$- \frac{1}{3} \cdot \frac{x (e^{u} \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

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

$- \frac{1}{3} \cdot \frac{x (e^{- x} \frac{d}{d x} [- x]) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate.

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Step 4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{1}{3} \cdot \frac{x (e^{- x} (- \frac{d}{d x} [x])) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.2

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

$- \frac{1}{3} \cdot \frac{x (e^{- x} (- 1 \cdot 1)) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $- 1$ by $1$ .

$- \frac{1}{3} \cdot \frac{x (e^{- x} \cdot - 1) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.3.2

Move $- 1$ to the left of $e^{- x}$ .

$- \frac{1}{3} \cdot \frac{x (- 1 \cdot e^{- x}) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.3.3

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$- \frac{1}{3} \cdot \frac{x (- e^{- x}) - e^{- x} \frac{d}{d x} [x]}{x^{2}}$

Step 4.4

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

$- \frac{1}{3} \cdot \frac{x (- e^{- x}) - e^{- x} \cdot 1}{x^{2}}$

Step 4.5

Combine fractions.

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Step 4.5.1

Multiply $- 1$ by $1$ .

$- \frac{1}{3} \cdot \frac{x (- e^{- x}) - e^{- x}}{x^{2}}$

Step 4.5.2

Multiply $\frac{x (- e^{- x}) - e^{- x}}{x^{2}}$ by $\frac{1}{3}$ .

$- \frac{x (- e^{- x}) - e^{- x}}{x^{2} \cdot 3}$

Step 4.5.3

Move $3$ to the left of $x^{2}$ .

$- \frac{x (- e^{- x}) - e^{- x}}{3 \cdot x^{2}}$

Step 5

Simplify.

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Step 5.1

Rewrite using the commutative property of multiplication.

$- \frac{- x e^{- x} - e^{- x}}{3 x^{2}}$

Step 5.2

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

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Step 5.2.1

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

$- \frac{e^{- x} (- x) - e^{- x}}{3 x^{2}}$

Step 5.2.2

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

$- \frac{e^{- x} (- x) + e^{- x} \cdot - 1}{3 x^{2}}$

Step 5.2.3

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

$- \frac{e^{- x} (- x - 1)}{3 x^{2}}$

Step 5.3

Factor $- 1$ out of $- x$ .

$- \frac{e^{- x} (- (x) - 1)}{3 x^{2}}$

Step 5.4

Rewrite $- 1$ as $- 1 (1)$ .

$- \frac{e^{- x} (- (x) - 1 (1))}{3 x^{2}}$

Step 5.5

Factor $- 1$ out of $- (x) - 1 (1)$ .

$- \frac{e^{- x} (- (x + 1))}{3 x^{2}}$

Step 5.6

Rewrite $- (x + 1)$ as $- 1 (x + 1)$ .

$- \frac{e^{- x} (- 1 (x + 1))}{3 x^{2}}$

Step 5.7

Move the negative in front of the fraction.

$- - \frac{e^{- x} (x + 1)}{3 x^{2}}$

Step 5.8

Multiply $- 1$ by $- 1$ .

$1 \frac{e^{- x} (x + 1)}{3 x^{2}}$

Step 5.9

Multiply $\frac{e^{- x} (x + 1)}{3 x^{2}}$ by $1$ .

$\frac{e^{- x} (x + 1)}{3 x^{2}}$