Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

$\frac{d}{d x} [\frac{3}{(x - 2) (x^{2} + 2)}]$

Step 1.2

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{(x - 2) (x^{2} + 2)}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{(x - 2) (x^{2} + 2)}]$ .

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

Step 1.3

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$3 (- {((x - 2) (x^{2} + 2))}^{- 2} \frac{d}{d x} [(x - 2) (x^{2} + 2)])$

Step 3

Multiply $- 1$ by $3$ .

$- 3 ({((x - 2) (x^{2} + 2))}^{- 2} \frac{d}{d x} [(x - 2) (x^{2} + 2)])$

Step 4

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} ((x - 2) \frac{d}{d x} [x^{2} + 2] + (x^{2} + 2) \frac{d}{d x} [x - 2])$

Step 5

Differentiate.

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Step 5.1

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} ((x - 2) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]) + (x^{2} + 2) \frac{d}{d x} [x - 2])$

Step 5.2

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} ((x - 2) (2 x + \frac{d}{d x} [2]) + (x^{2} + 2) \frac{d}{d x} [x - 2])$

Step 5.3

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} ((x - 2) (2 x + 0) + (x^{2} + 2) \frac{d}{d x} [x - 2])$

Step 5.4

Simplify the expression.

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Step 5.4.1

Add $2 x$ and $0$ .

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} ((x - 2) (2 x) + (x^{2} + 2) \frac{d}{d x} [x - 2])$

Step 5.4.2

Move $2$ to the left of $x - 2$ .

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

Step 5.5

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} (2 (x - 2) x + (x^{2} + 2) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]))$

Step 5.6

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

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

Step 5.7

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} (2 (x - 2) x + (x^{2} + 2) (1 + 0))$

Step 5.8

Simplify the expression.

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Step 5.8.1

Add $1$ and $0$ .

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

Step 5.8.2

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

$- 3 {((x - 2) (x^{2} + 2))}^{- 2} (2 (x - 2) x + x^{2} + 2)$

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Apply the product rule to $(x - 2) (x^{2} + 2)$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Combine terms.

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Step 6.5.1

Combine $- 3$ and $\frac{1}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$ .

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

Step 6.5.2

Move the negative in front of the fraction.

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

Step 6.5.3

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

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

Step 6.5.4

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

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

Step 6.5.5

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

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

Step 6.5.6

Add $1$ and $1$ .

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

Step 6.5.7

Multiply $2$ by $- 2$ .

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

Step 6.5.8

Add $2 x^{2}$ and $x^{2}$ .

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

Step 6.6

Reorder the factors of $- \frac{3}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}} (3 x^{2} - 4 x + 2)$ .

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

Step 6.7

Apply the distributive property.

$(- (3 x^{2}) - (- 4 x) - 1 \cdot 2) \frac{3}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$

Step 6.8

Simplify.

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Step 6.8.1

Multiply $3$ by $- 1$ .

$(- 3 x^{2} - (- 4 x) - 1 \cdot 2) \frac{3}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$

Step 6.8.2

Multiply $- 4$ by $- 1$ .

$(- 3 x^{2} + 4 x - 1 \cdot 2) \frac{3}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$

Step 6.8.3

Multiply $- 1$ by $2$ .

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

Step 6.9

Multiply $- 3 x^{2} + 4 x - 2$ by $\frac{3}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$ .

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

Step 6.10

Move $3$ to the left of $- 3 x^{2} + 4 x - 2$ .

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

Step 6.11

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

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

Step 6.12

Factor $- 1$ out of $4 x$ .

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

Step 6.13

Factor $- 1$ out of $- (3 x^{2}) - (- 4 x)$ .

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

Step 6.14

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

$\frac{3 (- (3 x^{2} - 4 x) - 1 (2))}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$

Step 6.15

Factor $- 1$ out of $- (3 x^{2} - 4 x) - 1 (2)$ .

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

Step 6.16

Rewrite $- (3 x^{2} - 4 x + 2)$ as $- 1 (3 x^{2} - 4 x + 2)$ .

$\frac{3 (- 1 (3 x^{2} - 4 x + 2))}{{(x - 2)}^{2} {(x^{2} + 2)}^{2}}$

Step 6.17

Move the negative in front of the fraction.

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