Calculus Examples

$\frac{1}{x (x + 1)}$

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x (x + 1)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x (x + 1)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x (x + 1)$ .

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x + 1$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$- {(x (x + 1))}^{- 2} (x (1 + 0) + (x + 1) \frac{d}{d x} [x])$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x$ by $1$ .

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

Step 4.5

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

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

Step 4.6

Simplify by adding terms.

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Step 4.6.1

Multiply $x + 1$ by $1$ .

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

Step 4.6.2

Add $x$ and $x$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Apply the product rule to $x (x + 1)$ .

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

Step 5.3

Reorder the factors of $- \frac{1}{x^{2} {(x + 1)}^{2}} (2 x + 1)$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $2$ by $- 1$ .

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

Step 5.6

Multiply $- 1$ by $1$ .

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

Move the negative in front of the fraction.

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