Calculus Examples

$\frac{10 x^{3} + 19 x^{2} - 5}{5 x + 2}$

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 $f (x) = 10 x^{3} + 19 x^{2} - 5$ and $g (x) = 5 x + 2$ .

$\frac{(5 x + 2) \frac{d}{d x} [10 x^{3} + 19 x^{2} - 5] - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 2

By the Sum Rule, the derivative of $10 x^{3} + 19 x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [19 x^{2}] + \frac{d}{d x} [- 5]$ .

$\frac{(5 x + 2) (\frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [19 x^{2}] + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 3

Evaluate $\frac{d}{d x} [10 x^{3}]$ .

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$\frac{(5 x + 2) (10 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [19 x^{2}] + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 3.2

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

$\frac{(5 x + 2) (10 (3 x^{2}) + \frac{d}{d x} [19 x^{2}] + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 3.3

Multiply $3$ by $10$ .

$\frac{(5 x + 2) (30 x^{2} + \frac{d}{d x} [19 x^{2}] + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4

Evaluate $\frac{d}{d x} [19 x^{2}]$ .

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Step 4.1

Since $19$ is constant with respect to $x$ , the derivative of $19 x^{2}$ with respect to $x$ is $19 \frac{d}{d x} [x^{2}]$ .

$\frac{(5 x + 2) (30 x^{2} + 19 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{(5 x + 2) (30 x^{2} + 19 (2 x) + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 4.3

Multiply $2$ by $19$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x + \frac{d}{d x} [- 5]) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 5

Differentiate.

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Step 5.1

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

$\frac{(5 x + 2) (30 x^{2} + 38 x + 0) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 5.2

Add $30 x^{2} + 38 x$ and $0$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) \frac{d}{d x} [5 x + 2]}{{(5 x + 2)}^{2}}$

Step 5.3

By the Sum Rule, the derivative of $5 x + 2$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [2]$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) (\frac{d}{d x} [5 x] + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 6

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 6.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) (5 \frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 6.2

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

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) (5 \cdot 1 + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 6.3

Multiply $5$ by $1$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) (5 + \frac{d}{d x} [2])}{{(5 x + 2)}^{2}}$

Step 7

Differentiate using the Constant Rule.

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Step 7.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) (5 + 0)}{{(5 x + 2)}^{2}}$

Step 7.2

Add $5$ and $0$ .

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3} + 19 x^{2} - 5) \cdot 5}{{(5 x + 2)}^{2}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{(5 x + 2) (30 x^{2} + 38 x) + (- (10 x^{3}) - (19 x^{2}) - - 5) \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.2

Apply the distributive property.

$\frac{(5 x + 2) (30 x^{2} + 38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Expand $(5 x + 2) (30 x^{2} + 38 x)$ using the FOIL Method.

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Step 8.3.1.1.1

Apply the distributive property.

$\frac{5 x (30 x^{2} + 38 x) + 2 (30 x^{2} + 38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.1.2

Apply the distributive property.

$\frac{5 x (30 x^{2}) + 5 x (38 x) + 2 (30 x^{2} + 38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.1.3

Apply the distributive property.

$\frac{5 x (30 x^{2}) + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2

Simplify and combine like terms.

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Step 8.3.1.2.1

Simplify each term.

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Step 8.3.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot 30 x \cdot x^{2} + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.2

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

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Step 8.3.1.2.1.2.1

Move $x^{2}$ .

$\frac{5 \cdot 30 (x^{2} x) + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.2.2

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

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Step 8.3.1.2.1.2.2.1

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

$\frac{5 \cdot 30 (x^{2} x^{1}) + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.2.2.2

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

$\frac{5 \cdot 30 x^{2 + 1} + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.2.3

Add $2$ and $1$ .

$\frac{5 \cdot 30 x^{3} + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.3

Multiply $5$ by $30$ .

$\frac{150 x^{3} + 5 x (38 x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.4

Rewrite using the commutative property of multiplication.

$\frac{150 x^{3} + 5 \cdot 38 x \cdot x + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 8.3.1.2.1.5.1

Move $x$ .

$\frac{150 x^{3} + 5 \cdot 38 (x \cdot x) + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.5.2

Multiply $x$ by $x$ .

$\frac{150 x^{3} + 5 \cdot 38 x^{2} + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.6

Multiply $5$ by $38$ .

$\frac{150 x^{3} + 190 x^{2} + 2 (30 x^{2}) + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.7

Multiply $30$ by $2$ .

$\frac{150 x^{3} + 190 x^{2} + 60 x^{2} + 2 (38 x) - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.1.8

Multiply $38$ by $2$ .

$\frac{150 x^{3} + 190 x^{2} + 60 x^{2} + 76 x - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.2.2

Add $190 x^{2}$ and $60 x^{2}$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - (10 x^{3}) \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.3

Multiply $10$ by $- 1$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 10 x^{3} \cdot 5 - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.4

Multiply $5$ by $- 10$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 50 x^{3} - (19 x^{2}) \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.5

Multiply $19$ by $- 1$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 50 x^{3} - 19 x^{2} \cdot 5 - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.6

Multiply $5$ by $- 19$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 50 x^{3} - 95 x^{2} - - 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.7

Multiply $- - 5 \cdot 5$ .

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Step 8.3.1.7.1

Multiply $- 1$ by $- 5$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 50 x^{3} - 95 x^{2} + 5 \cdot 5}{{(5 x + 2)}^{2}}$

Step 8.3.1.7.2

Multiply $5$ by $5$ .

$\frac{150 x^{3} + 250 x^{2} + 76 x - 50 x^{3} - 95 x^{2} + 25}{{(5 x + 2)}^{2}}$

Step 8.3.2

Subtract $50 x^{3}$ from $150 x^{3}$ .

$\frac{100 x^{3} + 250 x^{2} + 76 x - 95 x^{2} + 25}{{(5 x + 2)}^{2}}$

Step 8.3.3

Subtract $95 x^{2}$ from $250 x^{2}$ .

$\frac{100 x^{3} + 155 x^{2} + 76 x + 25}{{(5 x + 2)}^{2}}$