Calculus Examples

$y = \frac{e^{- x}}{x + 1}$

Step 1

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 $g (x) = x + 1$ .

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

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 $g (x) = - x$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $- 1$ by $1$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 3.5

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

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

Step 3.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{- (x + 1) e^{- x} - e^{- x} (1 + 0)}{{(x + 1)}^{2}}$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $1$ and $0$ .

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

Step 3.7.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $- 1$ by $1$ .

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

Step 4.3.1.2

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

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

Step 4.3.2

Subtract $e^{- x}$ from $- e^{- x}$ .

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

Step 4.4

Reorder terms.

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.6

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

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

Step 4.7

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 4.8

Factor $- 1$ out of $- (x) - 1 (2)$ .

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

Step 4.9

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

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

Step 4.10

Move the negative in front of the fraction.

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