Calculus Examples

$\frac{x^{3} - 8 x}{x - 1}$

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 $f (x) = x^{3} - 8 x$ and $g (x) = x - 1$ .

$\frac{(x - 1) \frac{d}{d x} [x^{3} - 8 x] - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} - 8 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]$ .

$\frac{(x - 1) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x]) - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 2.2

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

$\frac{(x - 1) (3 x^{2} + \frac{d}{d x} [- 8 x]) - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$\frac{(x - 1) (3 x^{2} - 8 \frac{d}{d x} [x]) - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 3.2

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

$\frac{(x - 1) (3 x^{2} - 8 \cdot 1) - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 3.3

Multiply $- 8$ by $1$ .

$\frac{(x - 1) (3 x^{2} - 8) - (x^{3} - 8 x) \frac{d}{d x} [x - 1]}{{(x - 1)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{(x - 1) (3 x^{2} - 8) - (x^{3} - 8 x) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 4.2

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

$\frac{(x - 1) (3 x^{2} - 8) - (x^{3} - 8 x) (1 + \frac{d}{d x} [- 1])}{{(x - 1)}^{2}}$

Step 4.3

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

$\frac{(x - 1) (3 x^{2} - 8) - (x^{3} - 8 x) (1 + 0)}{{(x - 1)}^{2}}$

Step 4.4

Add $1$ and $0$ .

$\frac{(x - 1) (3 x^{2} - 8) - (x^{3} - 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{(x - 1) (3 x^{2} - 8) + (- x^{3} - (- 8 x)) \cdot 1}{{(x - 1)}^{2}}$

Step 5.2

Apply the distributive property.

$\frac{(x - 1) (3 x^{2} - 8) - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Expand $(x - 1) (3 x^{2} - 8)$ using the FOIL Method.

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Step 5.3.1.1.1

Apply the distributive property.

$\frac{x (3 x^{2} - 8) - 1 (3 x^{2} - 8) - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.1.2

Apply the distributive property.

$\frac{x (3 x^{2}) + x \cdot - 8 - 1 (3 x^{2} - 8) - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.1.3

Apply the distributive property.

$\frac{x (3 x^{2}) + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2

Simplify each term.

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Step 5.3.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 x \cdot x^{2} + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.2

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

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Step 5.3.1.2.2.1

Move $x^{2}$ .

$\frac{3 (x^{2} x) + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.2.2

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

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Step 5.3.1.2.2.2.1

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

$\frac{3 (x^{2} x^{1}) + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.2.2.2

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

$\frac{3 x^{2 + 1} + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.2.3

Add $2$ and $1$ .

$\frac{3 x^{3} + x \cdot - 8 - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.3

Move $- 8$ to the left of $x$ .

$\frac{3 x^{3} - 8 \cdot x - 1 (3 x^{2}) - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.4

Multiply $3$ by $- 1$ .

$\frac{3 x^{3} - 8 x - 3 x^{2} - 1 \cdot - 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.2.5

Multiply $- 1$ by $- 8$ .

$\frac{3 x^{3} - 8 x - 3 x^{2} + 8 - x^{3} \cdot 1 - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.3

Multiply $- 1$ by $1$ .

$\frac{3 x^{3} - 8 x - 3 x^{2} + 8 - x^{3} - (- 8 x) \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.4

Multiply $- 8$ by $- 1$ .

$\frac{3 x^{3} - 8 x - 3 x^{2} + 8 - x^{3} + 8 x \cdot 1}{{(x - 1)}^{2}}$

Step 5.3.1.5

Multiply $8$ by $1$ .

$\frac{3 x^{3} - 8 x - 3 x^{2} + 8 - x^{3} + 8 x}{{(x - 1)}^{2}}$

Step 5.3.2

Combine the opposite terms in $3 x^{3} - 8 x - 3 x^{2} + 8 - x^{3} + 8 x$ .

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Step 5.3.2.1

Add $- 8 x$ and $8 x$ .

$\frac{3 x^{3} - 3 x^{2} + 8 - x^{3} + 0}{{(x - 1)}^{2}}$

Step 5.3.2.2

Add $3 x^{3} - 3 x^{2} + 8 - x^{3}$ and $0$ .

$\frac{3 x^{3} - 3 x^{2} + 8 - x^{3}}{{(x - 1)}^{2}}$

Step 5.3.3

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

$\frac{2 x^{3} - 3 x^{2} + 8}{{(x - 1)}^{2}}$

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