Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

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

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

Step 2.4.2

Move $3$ to the left of $2 x^{3} + 1$ .

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

Step 2.5

By the Sum Rule, the derivative of $2 x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [1]$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Multiply $3$ by $2$ .

$\frac{3 (2 x^{3} + 1) x^{2} - (x^{3} - 1) (6 x^{2} + \frac{d}{d x} [1])}{{(2 x^{3} + 1)}^{2}}$

Step 2.9

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

$\frac{3 (2 x^{3} + 1) x^{2} - (x^{3} - 1) (6 x^{2} + 0)}{{(2 x^{3} + 1)}^{2}}$

Step 2.10

Simplify the expression.

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Step 2.10.1

Add $6 x^{2}$ and $0$ .

$\frac{3 (2 x^{3} + 1) x^{2} - (x^{3} - 1) (6 x^{2})}{{(2 x^{3} + 1)}^{2}}$

Step 2.10.2

Multiply $6$ by $- 1$ .

$\frac{3 (2 x^{3} + 1) x^{2} - 6 (x^{3} - 1) x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(3 (2 x^{3}) + 3 \cdot 1) x^{2} - 6 (x^{3} - 1) x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{3 (2 x^{3}) x^{2} + 3 \cdot 1 x^{2} - 6 (x^{3} - 1) x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{3 (2 x^{3}) x^{2} + 3 \cdot 1 x^{2} + (- 6 x^{3} - 6 \cdot - 1) x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{3 (2 x^{3}) x^{2} + 3 \cdot 1 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.5.1.1.1

Move $x^{2}$ .

$\frac{3 (2 (x^{2} x^{3})) + 3 \cdot 1 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.1.2

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

$\frac{3 (2 x^{2 + 3}) + 3 \cdot 1 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.1.3

Add $2$ and $3$ .

$\frac{3 (2 x^{5}) + 3 \cdot 1 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.2

Multiply $2$ by $3$ .

$\frac{6 x^{5} + 3 \cdot 1 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.3

Multiply $3$ by $1$ .

$\frac{6 x^{5} + 3 x^{2} - 6 x^{3} x^{2} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.4

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.5.1.4.1

Move $x^{2}$ .

$\frac{6 x^{5} + 3 x^{2} - 6 (x^{2} x^{3}) - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.4.2

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

$\frac{6 x^{5} + 3 x^{2} - 6 x^{2 + 3} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.4.3

Add $2$ and $3$ .

$\frac{6 x^{5} + 3 x^{2} - 6 x^{5} - 6 \cdot - 1 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.1.5

Multiply $- 6$ by $- 1$ .

$\frac{6 x^{5} + 3 x^{2} - 6 x^{5} + 6 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.2

Combine the opposite terms in $6 x^{5} + 3 x^{2} - 6 x^{5} + 6 x^{2}$ .

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Step 3.5.2.1

Subtract $6 x^{5}$ from $6 x^{5}$ .

$\frac{3 x^{2} + 0 + 6 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.2.2

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

$\frac{3 x^{2} + 6 x^{2}}{{(2 x^{3} + 1)}^{2}}$

Step 3.5.3

Add $3 x^{2}$ and $6 x^{2}$ .

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