Calculus Examples

$y = \frac{x + 5}{x^{3} + x - 4}$

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 $f (x) = x + 5$ and $g (x) = x^{3} + x - 4$ .

$\frac{(x^{3} + x - 4) \frac{d}{d x} [x + 5] - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

$\frac{(x^{3} + x - 4) (\frac{d}{d x} [x] + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2.2

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

$\frac{(x^{3} + x - 4) (1 + \frac{d}{d x} [5]) - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2.3

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

$\frac{(x^{3} + x - 4) (1 + 0) - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

$\frac{(x^{3} + x - 4) \cdot 1 - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2.4.2

Multiply $x^{3} + x - 4$ by $1$ .

$\frac{x^{3} + x - 4 - (x + 5) \frac{d}{d x} [x^{3} + x - 4]}{{(x^{3} + x - 4)}^{2}}$

Step 2.5

By the Sum Rule, the derivative of $x^{3} + x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{x^{3} + x - 4 - (x + 5) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(x^{3} + x - 4)}^{2}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{x^{3} + x - 4 - (x + 5) (3 x^{2} + \frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(x^{3} + x - 4)}^{2}}$

Step 2.7

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

$\frac{x^{3} + x - 4 - (x + 5) (3 x^{2} + 1 + \frac{d}{d x} [- 4])}{{(x^{3} + x - 4)}^{2}}$

Step 2.8

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{x^{3} + x - 4 - (x + 5) (3 x^{2} + 1 + 0)}{{(x^{3} + x - 4)}^{2}}$

Step 2.9

Add $3 x^{2} + 1$ and $0$ .

$\frac{x^{3} + x - 4 - (x + 5) (3 x^{2} + 1)}{{(x^{3} + x - 4)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{x^{3} + x - 4 + (- x - 1 \cdot 5) (3 x^{2} + 1)}{{(x^{3} + x - 4)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 1$ by $5$ .

$\frac{x^{3} + x - 4 + (- x - 5) (3 x^{2} + 1)}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.2

Expand $(- x - 5) (3 x^{2} + 1)$ using the FOIL Method.

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Step 3.2.1.2.1

Apply the distributive property.

$\frac{x^{3} + x - 4 - x (3 x^{2} + 1) - 5 (3 x^{2} + 1)}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.2.2

Apply the distributive property.

$\frac{x^{3} + x - 4 - x (3 x^{2}) - x \cdot 1 - 5 (3 x^{2} + 1)}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.2.3

Apply the distributive property.

$\frac{x^{3} + x - 4 - x (3 x^{2}) - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3

Simplify each term.

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Step 3.2.1.3.1

Rewrite using the commutative property of multiplication.

$\frac{x^{3} + x - 4 - 1 \cdot 3 x \cdot x^{2} - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.2

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

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Step 3.2.1.3.2.1

Move $x^{2}$ .

$\frac{x^{3} + x - 4 - 1 \cdot 3 (x^{2} x) - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.2.2

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

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Step 3.2.1.3.2.2.1

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

$\frac{x^{3} + x - 4 - 1 \cdot 3 (x^{2} x^{1}) - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.2.2.2

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

$\frac{x^{3} + x - 4 - 1 \cdot 3 x^{2 + 1} - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.2.3

Add $2$ and $1$ .

$\frac{x^{3} + x - 4 - 1 \cdot 3 x^{3} - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.3

Multiply $- 1$ by $3$ .

$\frac{x^{3} + x - 4 - 3 x^{3} - x \cdot 1 - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.4

Multiply $- 1$ by $1$ .

$\frac{x^{3} + x - 4 - 3 x^{3} - x - 5 (3 x^{2}) - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.5

Multiply $3$ by $- 5$ .

$\frac{x^{3} + x - 4 - 3 x^{3} - x - 15 x^{2} - 5 \cdot 1}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.1.3.6

Multiply $- 5$ by $1$ .

$\frac{x^{3} + x - 4 - 3 x^{3} - x - 15 x^{2} - 5}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.2

Combine the opposite terms in $x^{3} + x - 4 - 3 x^{3} - x - 15 x^{2} - 5$ .

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Step 3.2.2.1

Subtract $x$ from $x$ .

$\frac{x^{3} - 4 - 3 x^{3} + 0 - 15 x^{2} - 5}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.2.2

Add $x^{3} - 4 - 3 x^{3}$ and $0$ .

$\frac{x^{3} - 4 - 3 x^{3} - 15 x^{2} - 5}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.3

Subtract $3 x^{3}$ from $x^{3}$ .

$\frac{- 2 x^{3} - 4 - 15 x^{2} - 5}{{(x^{3} + x - 4)}^{2}}$

Step 3.2.4

Subtract $5$ from $- 4$ .

$\frac{- 2 x^{3} - 15 x^{2} - 9}{{(x^{3} + x - 4)}^{2}}$

Step 3.3

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

$\frac{- (2 x^{3}) - 15 x^{2} - 9}{{(x^{3} + x - 4)}^{2}}$

Step 3.4

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

$\frac{- (2 x^{3}) - (15 x^{2}) - 9}{{(x^{3} + x - 4)}^{2}}$

Step 3.5

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

$\frac{- (2 x^{3} + 15 x^{2}) - 9}{{(x^{3} + x - 4)}^{2}}$

Step 3.6

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (2 x^{3} + 15 x^{2}) - 1 (9)}{{(x^{3} + x - 4)}^{2}}$

Step 3.7

Factor $- 1$ out of $- (2 x^{3} + 15 x^{2}) - 1 (9)$ .

$\frac{- (2 x^{3} + 15 x^{2} + 9)}{{(x^{3} + x - 4)}^{2}}$

Step 3.8

Rewrite $- (2 x^{3} + 15 x^{2} + 9)$ as $- 1 (2 x^{3} + 15 x^{2} + 9)$ .

$\frac{- 1 (2 x^{3} + 15 x^{2} + 9)}{{(x^{3} + x - 4)}^{2}}$

Step 3.9

Move the negative in front of the fraction.

$- \frac{2 x^{3} + 15 x^{2} + 9}{{(x^{3} + x - 4)}^{2}}$