Calculus Examples

${(\frac{x + 5}{x^{2} + 2})}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \frac{x + 5}{x^{2} + 2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x + 5}{x^{2} + 2}$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [\frac{x + 5}{x^{2} + 2}]$

Step 1.2

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

$2 u \frac{d}{d x} [\frac{x + 5}{x^{2} + 2}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x + 5}{x^{2} + 2}$ .

$2 \frac{x + 5}{x^{2} + 2} \frac{d}{d x} [\frac{x + 5}{x^{2} + 2}]$

Step 2

Combine $2$ and $\frac{x + 5}{x^{2} + 2}$ .

$\frac{2 (x + 5)}{x^{2} + 2} \frac{d}{d x} [\frac{x + 5}{x^{2} + 2}]$

Step 3

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 $f (x) = x + 5$ and $g (x) = x^{2} + 2$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{(x^{2} + 2) \frac{d}{d x} [x + 5] - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{(x^{2} + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4.2

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

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{(x^{2} + 2) (1 + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{(x^{2} + 2) (1 + 0) - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{(x^{2} + 2) \cdot 1 - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4.4.2

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

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - (x + 5) \frac{d}{d x} [x^{2} + 2]}{{(x^{2} + 2)}^{2}}$

Step 4.5

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

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - (x + 5) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])}{{(x^{2} + 2)}^{2}}$

Step 4.6

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

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - (x + 5) (2 x + \frac{d}{d x} [2])}{{(x^{2} + 2)}^{2}}$

Step 4.7

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - (x + 5) (2 x + 0)}{{(x^{2} + 2)}^{2}}$

Step 4.8

Combine fractions.

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Step 4.8.1

Add $2 x$ and $0$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - (x + 5) (2 x)}{{(x^{2} + 2)}^{2}}$

Step 4.8.2

Multiply $2$ by $- 1$ .

$\frac{2 (x + 5)}{x^{2} + 2} \cdot \frac{x^{2} + 2 - 2 (x + 5) x}{{(x^{2} + 2)}^{2}}$

Step 4.8.3

Multiply $\frac{2 (x + 5)}{x^{2} + 2}$ by $\frac{x^{2} + 2 - 2 (x + 5) x}{{(x^{2} + 2)}^{2}}$ .

$\frac{2 (x + 5) (x^{2} + 2 - 2 (x + 5) x)}{(x^{2} + 2) {(x^{2} + 2)}^{2}}$

Step 5

Multiply $x^{2} + 2$ by ${(x^{2} + 2)}^{2}$ by adding the exponents.

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Step 5.1

Multiply $x^{2} + 2$ by ${(x^{2} + 2)}^{2}$ .

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Step 5.1.1

Raise $x^{2} + 2$ to the power of $1$ .

$\frac{2 (x + 5) (x^{2} + 2 - 2 (x + 5) x)}{{(x^{2} + 2)}^{1} {(x^{2} + 2)}^{2}}$

Step 5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 (x + 5) (x^{2} + 2 - 2 (x + 5) x)}{{(x^{2} + 2)}^{1 + 2}}$

Step 5.2

Add $1$ and $2$ .

$\frac{2 (x + 5) (x^{2} + 2 - 2 (x + 5) x)}{{(x^{2} + 2)}^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot 5) (x^{2} + 2 - 2 (x + 5) x)}{{(x^{2} + 2)}^{3}}$

Step 6.2

Apply the distributive property.

$\frac{(2 x + 2 \cdot 5) (x^{2} + 2 + (- 2 x - 2 \cdot 5) x)}{{(x^{2} + 2)}^{3}}$

Step 6.3

Apply the distributive property.

$\frac{(2 x + 2 \cdot 5) (x^{2} + 2 - 2 x \cdot x - 2 \cdot 5 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $2$ by $5$ .

$\frac{(2 x + 10) (x^{2} + 2 - 2 x \cdot x - 2 \cdot 5 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.2

Simplify each term.

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Step 6.4.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.2.1.1

Move $x$ .

$\frac{(2 x + 10) (x^{2} + 2 - 2 (x \cdot x) - 2 \cdot 5 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.2.1.2

Multiply $x$ by $x$ .

$\frac{(2 x + 10) (x^{2} + 2 - 2 x^{2} - 2 \cdot 5 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.2.2

Multiply $- 2$ by $5$ .

$\frac{(2 x + 10) (x^{2} + 2 - 2 x^{2} - 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.3

Subtract $2 x^{2}$ from $x^{2}$ .

$\frac{(2 x + 10) (- x^{2} + 2 - 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.4

Expand $(2 x + 10) (- x^{2} + 2 - 10 x)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x (- x^{2}) + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5

Simplify each term.

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Step 6.4.5.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot - 1 x \cdot x^{2} + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.4.5.2.1

Move $x^{2}$ .

$\frac{2 \cdot - 1 (x^{2} x) + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.2.2

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

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Step 6.4.5.2.2.1

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

$\frac{2 \cdot - 1 (x^{2} x^{1}) + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \cdot - 1 x^{2 + 1} + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.2.3

Add $2$ and $1$ .

$\frac{2 \cdot - 1 x^{3} + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.3

Multiply $2$ by $- 1$ .

$\frac{- 2 x^{3} + 2 x \cdot 2 + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.4

Multiply $2$ by $2$ .

$\frac{- 2 x^{3} + 4 x + 2 x (- 10 x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.5

Rewrite using the commutative property of multiplication.

$\frac{- 2 x^{3} + 4 x + 2 \cdot - 10 x \cdot x + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.4.5.6.1

Move $x$ .

$\frac{- 2 x^{3} + 4 x + 2 \cdot - 10 (x \cdot x) + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.6.2

Multiply $x$ by $x$ .

$\frac{- 2 x^{3} + 4 x + 2 \cdot - 10 x^{2} + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.7

Multiply $2$ by $- 10$ .

$\frac{- 2 x^{3} + 4 x - 20 x^{2} + 10 (- x^{2}) + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.8

Multiply $- 1$ by $10$ .

$\frac{- 2 x^{3} + 4 x - 20 x^{2} - 10 x^{2} + 10 \cdot 2 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.9

Multiply $10$ by $2$ .

$\frac{- 2 x^{3} + 4 x - 20 x^{2} - 10 x^{2} + 20 + 10 (- 10 x)}{{(x^{2} + 2)}^{3}}$

Step 6.4.5.10

Multiply $- 10$ by $10$ .

$\frac{- 2 x^{3} + 4 x - 20 x^{2} - 10 x^{2} + 20 - 100 x}{{(x^{2} + 2)}^{3}}$

Step 6.4.6

Subtract $100 x$ from $4 x$ .

$\frac{- 2 x^{3} - 20 x^{2} - 10 x^{2} + 20 - 96 x}{{(x^{2} + 2)}^{3}}$

Step 6.4.7

Subtract $10 x^{2}$ from $- 20 x^{2}$ .

$\frac{- 2 x^{3} - 30 x^{2} + 20 - 96 x}{{(x^{2} + 2)}^{3}}$

Step 6.5

Reorder terms.

$\frac{- 2 x^{3} - 30 x^{2} - 96 x + 20}{{(x^{2} + 2)}^{3}}$

Step 6.6

Factor $2$ out of $- 2 x^{3} - 30 x^{2} - 96 x + 20$ .

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Step 6.6.1

Factor $2$ out of $- 2 x^{3}$ .

$\frac{2 (- x^{3}) - 30 x^{2} - 96 x + 20}{{(x^{2} + 2)}^{3}}$

Step 6.6.2

Factor $2$ out of $- 30 x^{2}$ .

$\frac{2 (- x^{3}) + 2 (- 15 x^{2}) - 96 x + 20}{{(x^{2} + 2)}^{3}}$

Step 6.6.3

Factor $2$ out of $- 96 x$ .

$\frac{2 (- x^{3}) + 2 (- 15 x^{2}) + 2 (- 48 x) + 20}{{(x^{2} + 2)}^{3}}$

Step 6.6.4

Factor $2$ out of $20$ .

$\frac{2 (- x^{3}) + 2 (- 15 x^{2}) + 2 (- 48 x) + 2 (10)}{{(x^{2} + 2)}^{3}}$

Step 6.6.5

Factor $2$ out of $2 (- x^{3}) + 2 (- 15 x^{2})$ .

$\frac{2 (- x^{3} - 15 x^{2}) + 2 (- 48 x) + 2 (10)}{{(x^{2} + 2)}^{3}}$

Step 6.6.6

Factor $2$ out of $2 (- x^{3} - 15 x^{2}) + 2 (- 48 x)$ .

$\frac{2 (- x^{3} - 15 x^{2} - 48 x) + 2 (10)}{{(x^{2} + 2)}^{3}}$

Step 6.6.7

Factor $2$ out of $2 (- x^{3} - 15 x^{2} - 48 x) + 2 (10)$ .

$\frac{2 (- x^{3} - 15 x^{2} - 48 x + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.7

Factor $- 1$ out of $- x^{3}$ .

$\frac{2 (- (x^{3}) - 15 x^{2} - 48 x + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.8

Factor $- 1$ out of $- 15 x^{2}$ .

$\frac{2 (- (x^{3}) - (15 x^{2}) - 48 x + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.9

Factor $- 1$ out of $- (x^{3}) - (15 x^{2})$ .

$\frac{2 (- (x^{3} + 15 x^{2}) - 48 x + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.10

Factor $- 1$ out of $- 48 x$ .

$\frac{2 (- (x^{3} + 15 x^{2}) - (48 x) + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.11

Factor $- 1$ out of $- (x^{3} + 15 x^{2}) - (48 x)$ .

$\frac{2 (- (x^{3} + 15 x^{2} + 48 x) + 10)}{{(x^{2} + 2)}^{3}}$

Step 6.12

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{2 (- (x^{3} + 15 x^{2} + 48 x) - 1 (- 10))}{{(x^{2} + 2)}^{3}}$

Step 6.13

Factor $- 1$ out of $- (x^{3} + 15 x^{2} + 48 x) - 1 (- 10)$ .

$\frac{2 (- (x^{3} + 15 x^{2} + 48 x - 10))}{{(x^{2} + 2)}^{3}}$

Step 6.14

Rewrite $- (x^{3} + 15 x^{2} + 48 x - 10)$ as $- 1 (x^{3} + 15 x^{2} + 48 x - 10)$ .

$\frac{2 (- 1 (x^{3} + 15 x^{2} + 48 x - 10))}{{(x^{2} + 2)}^{3}}$

Step 6.15

Move the negative in front of the fraction.

$- \frac{2 (x^{3} + 15 x^{2} + 48 x - 10)}{{(x^{2} + 2)}^{3}}$