Calculus Examples

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

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 $f (x) = 8 x^{3} - 1$ and $g (x) = 2 x - 1$ .

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

Step 2

By the Sum Rule, the derivative of $8 x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [- 1]$ .

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

Step 3

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

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

Step 3.2

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

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

Step 3.3

Multiply $3$ by $8$ .

$\frac{(2 x - 1) (24 x^{2} + \frac{d}{d x} [- 1]) - (8 x^{3} - 1) \frac{d}{d x} [2 x - 1]}{{(2 x - 1)}^{2}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{(2 x - 1) (24 x^{2} + 0) - (8 x^{3} - 1) \frac{d}{d x} [2 x - 1]}{{(2 x - 1)}^{2}}$

Step 4.2

Add $24 x^{2}$ and $0$ .

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) \frac{d}{d x} [2 x - 1]}{{(2 x - 1)}^{2}}$

Step 4.3

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])}{{(2 x - 1)}^{2}}$

Step 5

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 5.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])}{{(2 x - 1)}^{2}}$

Step 5.2

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

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) (2 \cdot 1 + \frac{d}{d x} [- 1])}{{(2 x - 1)}^{2}}$

Step 5.3

Multiply $2$ by $1$ .

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) (2 + \frac{d}{d x} [- 1])}{{(2 x - 1)}^{2}}$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

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

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) (2 + 0)}{{(2 x - 1)}^{2}}$

Step 6.2

Add $2$ and $0$ .

$\frac{(2 x - 1) (24 x^{2}) - (8 x^{3} - 1) \cdot 2}{{(2 x - 1)}^{2}}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{2 x (24 x^{2}) - 1 (24 x^{2}) - (8 x^{3} - 1) \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.2

Apply the distributive property.

$\frac{2 x (24 x^{2}) - 1 (24 x^{2}) + (- (8 x^{3}) - - 1) \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.3

Apply the distributive property.

$\frac{2 x (24 x^{2}) - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Simplify each term.

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Step 7.4.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 \cdot 24 x \cdot x^{2} - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.2

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

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Step 7.4.1.2.1

Move $x^{2}$ .

$\frac{2 \cdot 24 (x^{2} x) - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.2.2

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

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Step 7.4.1.2.2.1

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

$\frac{2 \cdot 24 (x^{2} x^{1}) - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.2.2.2

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

$\frac{2 \cdot 24 x^{2 + 1} - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.2.3

Add $2$ and $1$ .

$\frac{2 \cdot 24 x^{3} - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.3

Multiply $2$ by $24$ .

$\frac{48 x^{3} - 1 (24 x^{2}) - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.4

Multiply $24$ by $- 1$ .

$\frac{48 x^{3} - 24 x^{2} - (8 x^{3}) \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.5

Multiply $8$ by $- 1$ .

$\frac{48 x^{3} - 24 x^{2} - 8 x^{3} \cdot 2 - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.6

Multiply $2$ by $- 8$ .

$\frac{48 x^{3} - 24 x^{2} - 16 x^{3} - - 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.7

Multiply $- - 1 \cdot 2$ .

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Step 7.4.1.7.1

Multiply $- 1$ by $- 1$ .

$\frac{48 x^{3} - 24 x^{2} - 16 x^{3} + 1 \cdot 2}{{(2 x - 1)}^{2}}$

Step 7.4.1.7.2

Multiply $2$ by $1$ .

$\frac{48 x^{3} - 24 x^{2} - 16 x^{3} + 2}{{(2 x - 1)}^{2}}$

Step 7.4.2

Subtract $16 x^{3}$ from $48 x^{3}$ .

$\frac{32 x^{3} - 24 x^{2} + 2}{{(2 x - 1)}^{2}}$

Step 7.5

Factor $2$ out of $32 x^{3} - 24 x^{2} + 2$ .

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Step 7.5.1

Factor $2$ out of $32 x^{3}$ .

$\frac{2 (16 x^{3}) - 24 x^{2} + 2}{{(2 x - 1)}^{2}}$

Step 7.5.2

Factor $2$ out of $- 24 x^{2}$ .

$\frac{2 (16 x^{3}) + 2 (- 12 x^{2}) + 2}{{(2 x - 1)}^{2}}$

Step 7.5.3

Factor $2$ out of $2$ .

$\frac{2 (16 x^{3}) + 2 (- 12 x^{2}) + 2 (1)}{{(2 x - 1)}^{2}}$

Step 7.5.4

Factor $2$ out of $2 (16 x^{3}) + 2 (- 12 x^{2})$ .

$\frac{2 (16 x^{3} - 12 x^{2}) + 2 (1)}{{(2 x - 1)}^{2}}$

Step 7.5.5

Factor $2$ out of $2 (16 x^{3} - 12 x^{2}) + 2 (1)$ .

$\frac{2 (16 x^{3} - 12 x^{2} + 1)}{{(2 x - 1)}^{2}}$