Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $3$ by $2$ .

$\frac{(x + 2) (6 x^{2} + \frac{d}{d x} [- 5 x^{2}]) - (2 x^{3} - 5 x^{2}) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$\frac{(x + 2) (6 x^{2} - 5 \frac{d}{d x} [x^{2}]) - (2 x^{3} - 5 x^{2}) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.6

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

$\frac{(x + 2) (6 x^{2} - 5 (2 x)) - (2 x^{3} - 5 x^{2}) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.7

Multiply $2$ by $- 5$ .

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2}) \frac{d}{d x} [x + 2]}{{(x + 2)}^{2}}$

Step 2.8

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

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.9

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

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2}) (1 + \frac{d}{d x} [2])}{{(x + 2)}^{2}}$

Step 2.10

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

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2}) (1 + 0)}{{(x + 2)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2}) \cdot 1}{{(x + 2)}^{2}}$

Step 2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3} - 5 x^{2})}{{(x + 2)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x + 2) (6 x^{2} - 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x + 2) (6 x^{2} - 10 x)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (6 x^{2} - 10 x) + 2 (6 x^{2} - 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (6 x^{2}) + x (- 10 x) + 2 (6 x^{2} - 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (6 x^{2}) + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{6 x \cdot x^{2} + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x^{2}$ .

$\frac{6 (x^{2} x) + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.2.2

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

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Step 3.2.1.2.1.2.2.1

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

$\frac{6 (x^{2} x^{1}) + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.2.2.2

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

$\frac{6 x^{2 + 1} + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.2.3

Add $2$ and $1$ .

$\frac{6 x^{3} + x (- 10 x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.3

Rewrite using the commutative property of multiplication.

$\frac{6 x^{3} - 10 x \cdot x + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.4

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

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Step 3.2.1.2.1.4.1

Move $x$ .

$\frac{6 x^{3} - 10 (x \cdot x) + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.4.2

Multiply $x$ by $x$ .

$\frac{6 x^{3} - 10 x^{2} + 2 (6 x^{2}) + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.5

Multiply $6$ by $2$ .

$\frac{6 x^{3} - 10 x^{2} + 12 x^{2} + 2 (- 10 x) - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.1.6

Multiply $- 10$ by $2$ .

$\frac{6 x^{3} - 10 x^{2} + 12 x^{2} - 20 x - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.2.2

Add $- 10 x^{2}$ and $12 x^{2}$ .

$\frac{6 x^{3} + 2 x^{2} - 20 x - (2 x^{3}) - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.3

Multiply $2$ by $- 1$ .

$\frac{6 x^{3} + 2 x^{2} - 20 x - 2 x^{3} - (- 5 x^{2})}{{(x + 2)}^{2}}$

Step 3.2.1.4

Multiply $- 5$ by $- 1$ .

$\frac{6 x^{3} + 2 x^{2} - 20 x - 2 x^{3} + 5 x^{2}}{{(x + 2)}^{2}}$

Step 3.2.2

Subtract $2 x^{3}$ from $6 x^{3}$ .

$\frac{4 x^{3} + 2 x^{2} - 20 x + 5 x^{2}}{{(x + 2)}^{2}}$

Step 3.2.3

Add $2 x^{2}$ and $5 x^{2}$ .

$\frac{4 x^{3} + 7 x^{2} - 20 x}{{(x + 2)}^{2}}$

Step 3.3

Factor $x$ out of $4 x^{3} + 7 x^{2} - 20 x$ .

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Step 3.3.1

Factor $x$ out of $4 x^{3}$ .

$\frac{x (4 x^{2}) + 7 x^{2} - 20 x}{{(x + 2)}^{2}}$

Step 3.3.2

Factor $x$ out of $7 x^{2}$ .

$\frac{x (4 x^{2}) + x (7 x) - 20 x}{{(x + 2)}^{2}}$

Step 3.3.3

Factor $x$ out of $- 20 x$ .

$\frac{x (4 x^{2}) + x (7 x) + x \cdot - 20}{{(x + 2)}^{2}}$

Step 3.3.4

Factor $x$ out of $x (4 x^{2}) + x (7 x)$ .

$\frac{x (4 x^{2} + 7 x) + x \cdot - 20}{{(x + 2)}^{2}}$

Step 3.3.5

Factor $x$ out of $x (4 x^{2} + 7 x) + x \cdot - 20$ .

$\frac{x (4 x^{2} + 7 x - 20)}{{(x + 2)}^{2}}$