Calculus Examples

$\frac{1}{x^{2} + x}$

Step 1

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

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

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 $g (x) = x^{2} + x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} + x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x^{2} + x$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} + x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x]$ .

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

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $2$ by $- 1$ .

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

Step 4.5

Multiply $- 1$ by $1$ .

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

Step 4.6

Simplify the denominator.

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Step 4.6.1

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

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Step 4.6.1.1

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

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

Step 4.6.1.2

Raise $x$ to the power of $1$ .

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

Step 4.6.1.3

Factor $x$ out of $x^{1}$ .

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

Step 4.6.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 4.6.2

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

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

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

Move the negative in front of the fraction.

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