Calculus Examples

$y = \frac{5 + 3 x^{3}}{1 + 2 x^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{5 + 3 x^{3}}{1 + 2 x^{3}})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = 5 + 3 x^{3}$ and $g (x) = 1 + 2 x^{3}$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $5 + 3 x^{3}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [3 x^{3}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Add $0$ and $\frac{d}{d x} [3 x^{3}]$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.2.7

Multiply $3$ by $3$ .

$\frac{9 (1 + 2 x^{3}) x^{2} - (5 + 3 x^{3}) \frac{d}{d x} [1 + 2 x^{3}]}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.8

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

$\frac{9 (1 + 2 x^{3}) x^{2} - (5 + 3 x^{3}) (\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{3}])}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.9

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

$\frac{9 (1 + 2 x^{3}) x^{2} - (5 + 3 x^{3}) (0 + \frac{d}{d x} [2 x^{3}])}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.10

Add $0$ and $\frac{d}{d x} [2 x^{3}]$ .

$\frac{9 (1 + 2 x^{3}) x^{2} - (5 + 3 x^{3}) \frac{d}{d x} [2 x^{3}]}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.11

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

$\frac{9 (1 + 2 x^{3}) x^{2} - (5 + 3 x^{3}) (2 \frac{d}{d x} [x^{3}])}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.12

Multiply $2$ by $- 1$ .

$\frac{9 (1 + 2 x^{3}) x^{2} - 2 (5 + 3 x^{3}) \frac{d}{d x} [x^{3}]}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.13

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

$\frac{9 (1 + 2 x^{3}) x^{2} - 2 (5 + 3 x^{3}) (3 x^{2})}{{(1 + 2 x^{3})}^{2}}$

Step 3.2.14

Multiply $3$ by $- 2$ .

$\frac{9 (1 + 2 x^{3}) x^{2} - 6 (5 + 3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{(9 \cdot 1 + 9 (2 x^{3})) x^{2} - 6 (5 + 3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{9 \cdot 1 x^{2} + 9 (2 x^{3}) x^{2} - 6 (5 + 3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.3

Apply the distributive property.

$\frac{9 \cdot 1 x^{2} + 9 (2 x^{3}) x^{2} + (- 6 \cdot 5 - 6 (3 x^{3})) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.4

Apply the distributive property.

$\frac{9 \cdot 1 x^{2} + 9 (2 x^{3}) x^{2} - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Simplify each term.

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Step 3.3.5.1.1

Multiply $9$ by $1$ .

$\frac{9 x^{2} + 9 (2 x^{3}) x^{2} - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.2

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

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Step 3.3.5.1.2.1

Move $x^{2}$ .

$\frac{9 x^{2} + 9 (2 (x^{2} x^{3})) - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.2.2

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

$\frac{9 x^{2} + 9 (2 x^{2 + 3}) - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.2.3

Add $2$ and $3$ .

$\frac{9 x^{2} + 9 (2 x^{5}) - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.3

Multiply $2$ by $9$ .

$\frac{9 x^{2} + 18 x^{5} - 6 \cdot 5 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.4

Multiply $- 6$ by $5$ .

$\frac{9 x^{2} + 18 x^{5} - 30 x^{2} - 6 (3 x^{3}) x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.5

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

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Step 3.3.5.1.5.1

Move $x^{2}$ .

$\frac{9 x^{2} + 18 x^{5} - 30 x^{2} - 6 (3 (x^{2} x^{3}))}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.5.2

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

$\frac{9 x^{2} + 18 x^{5} - 30 x^{2} - 6 (3 x^{2 + 3})}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.5.3

Add $2$ and $3$ .

$\frac{9 x^{2} + 18 x^{5} - 30 x^{2} - 6 (3 x^{5})}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.1.6

Multiply $3$ by $- 6$ .

$\frac{9 x^{2} + 18 x^{5} - 30 x^{2} - 18 x^{5}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.2

Combine the opposite terms in $9 x^{2} + 18 x^{5} - 30 x^{2} - 18 x^{5}$ .

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Step 3.3.5.2.1

Subtract $18 x^{5}$ from $18 x^{5}$ .

$\frac{9 x^{2} - 30 x^{2} + 0}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.2.2

Add $9 x^{2} - 30 x^{2}$ and $0$ .

$\frac{9 x^{2} - 30 x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.5.3

Subtract $30 x^{2}$ from $9 x^{2}$ .

$\frac{- 21 x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 3.3.6

Move the negative in front of the fraction.

$- \frac{21 x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{21 x^{2}}{{(1 + 2 x^{3})}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{21 x^{2}}{{(1 + 2 x^{3})}^{2}}$