Calculus Examples

$y = (2 x^{3} - 4 x^{2}) (3 x^{5} + x^{2})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 2 x^{3} - 4 x^{2}$ and $g (x) = 3 x^{5} + x^{2}$ .

$(2 x^{3} - 4 x^{2}) \frac{d}{d x} [3 x^{5} + x^{2}] + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2

Differentiate.

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Step 2.1

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

$(2 x^{3} - 4 x^{2}) (\frac{d}{d x} [3 x^{5}] + \frac{d}{d x} [x^{2}]) + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{5}$ with respect to $x$ is $3 \frac{d}{d x} [x^{5}]$ .

$(2 x^{3} - 4 x^{2}) (3 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [x^{2}]) + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$(2 x^{3} - 4 x^{2}) (3 (5 x^{4}) + \frac{d}{d x} [x^{2}]) + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2.4

Multiply $5$ by $3$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + \frac{d}{d x} [x^{2}]) + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2.5

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

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) \frac{d}{d x} [2 x^{3} - 4 x^{2}]$

Step 2.6

By the Sum Rule, the derivative of $2 x^{3} - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 4 x^{2}]$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 4 x^{2}])$

Step 2.7

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

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}])$

Step 2.8

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

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (2 (3 x^{2}) + \frac{d}{d x} [- 4 x^{2}])$

Step 2.9

Multiply $3$ by $2$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} + \frac{d}{d x} [- 4 x^{2}])$

Step 2.10

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 4 \frac{d}{d x} [x^{2}])$

Step 2.11

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

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 4 (2 x))$

Step 2.12

Multiply $2$ by $- 4$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3

Simplify.

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Step 3.1

Factor $1$ out of $(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$ .

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Step 3.1.1

Factor $1$ out of $(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x)$ .

$1 ((2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x)) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.1.2

Factor $1$ out of $(3 x^{5} + x^{2}) (6 x^{2} - 8 x)$ .

$1 ((2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x)) + 1 ((3 x^{5} + x^{2}) (6 x^{2} - 8 x))$

Step 3.1.3

Factor $1$ out of $1 ((2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x)) + 1 ((3 x^{5} + x^{2}) (6 x^{2} - 8 x))$ .

$1 ((2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x))$

Step 3.2

Multiply $(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$ by $1$ .

$(2 x^{3} - 4 x^{2}) (15 x^{4} + 2 x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.3

Reorder terms.

$(15 x^{4} + 2 x) (2 x^{3} - 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(15 x^{4} + 2 x) (2 x^{3} - 4 x^{2})$ using the FOIL Method.

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Step 3.4.1.1

Apply the distributive property.

$15 x^{4} (2 x^{3} - 4 x^{2}) + 2 x (2 x^{3} - 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.1.2

Apply the distributive property.

$15 x^{4} (2 x^{3}) + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3} - 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.1.3

Apply the distributive property.

$15 x^{4} (2 x^{3}) + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Rewrite using the commutative property of multiplication.

$15 \cdot 2 x^{4} x^{3} + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.2

Multiply $x^{4}$ by $x^{3}$ by adding the exponents.

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Step 3.4.2.2.1

Move $x^{3}$ .

$15 \cdot 2 (x^{3} x^{4}) + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.2.2

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

$15 \cdot 2 x^{3 + 4} + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.2.3

Add $3$ and $4$ .

$15 \cdot 2 x^{7} + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.3

Multiply $15$ by $2$ .

$30 x^{7} + 15 x^{4} (- 4 x^{2}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.4

Rewrite using the commutative property of multiplication.

$30 x^{7} + 15 \cdot - 4 x^{4} x^{2} + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.5

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 3.4.2.5.1

Move $x^{2}$ .

$30 x^{7} + 15 \cdot - 4 (x^{2} x^{4}) + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.5.2

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

$30 x^{7} + 15 \cdot - 4 x^{2 + 4} + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.5.3

Add $2$ and $4$ .

$30 x^{7} + 15 \cdot - 4 x^{6} + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.6

Multiply $15$ by $- 4$ .

$30 x^{7} - 60 x^{6} + 2 x (2 x^{3}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.7

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 2 \cdot 2 x \cdot x^{3} + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.8

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

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Step 3.4.2.8.1

Move $x^{3}$ .

$30 x^{7} - 60 x^{6} + 2 \cdot 2 (x^{3} x) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.8.2

Multiply $x^{3}$ by $x$ .

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Step 3.4.2.8.2.1

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

$30 x^{7} - 60 x^{6} + 2 \cdot 2 (x^{3} x^{1}) + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.8.2.2

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

$30 x^{7} - 60 x^{6} + 2 \cdot 2 x^{3 + 1} + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.8.3

Add $3$ and $1$ .

$30 x^{7} - 60 x^{6} + 2 \cdot 2 x^{4} + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.9

Multiply $2$ by $2$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 x (- 4 x^{2}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.10

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 \cdot - 4 x \cdot x^{2} + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.11

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

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Step 3.4.2.11.1

Move $x^{2}$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 \cdot - 4 (x^{2} x) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.11.2

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

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Step 3.4.2.11.2.1

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

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 \cdot - 4 (x^{2} x^{1}) + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.11.2.2

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

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 \cdot - 4 x^{2 + 1} + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.11.3

Add $2$ and $1$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} + 2 \cdot - 4 x^{3} + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.2.12

Multiply $2$ by $- 4$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + (3 x^{5} + x^{2}) (6 x^{2} - 8 x)$

Step 3.4.3

Expand $(3 x^{5} + x^{2}) (6 x^{2} - 8 x)$ using the FOIL Method.

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Step 3.4.3.1

Apply the distributive property.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 x^{5} (6 x^{2} - 8 x) + x^{2} (6 x^{2} - 8 x)$

Step 3.4.3.2

Apply the distributive property.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 x^{5} (6 x^{2}) + 3 x^{5} (- 8 x) + x^{2} (6 x^{2} - 8 x)$

Step 3.4.3.3

Apply the distributive property.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 x^{5} (6 x^{2}) + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4

Simplify each term.

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Step 3.4.4.1

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 \cdot 6 x^{5} x^{2} + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.2

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 3.4.4.2.1

Move $x^{2}$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 \cdot 6 (x^{2} x^{5}) + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.2.2

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 \cdot 6 x^{2 + 5} + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.2.3

Add $2$ and $5$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 3 \cdot 6 x^{7} + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.3

Multiply $3$ by $6$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 x^{5} (- 8 x) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.4

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 \cdot - 8 x^{5} x + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.5

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.4.4.5.1

Move $x$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 \cdot - 8 (x \cdot x^{5}) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.5.2

Multiply $x$ by $x^{5}$ .

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Step 3.4.4.5.2.1

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 \cdot - 8 (x^{1} x^{5}) + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.5.2.2

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 \cdot - 8 x^{1 + 5} + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.5.3

Add $1$ and $5$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} + 3 \cdot - 8 x^{6} + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.6

Multiply $3$ by $- 8$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + x^{2} (6 x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.7

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{2} x^{2} + x^{2} (- 8 x)$

Step 3.4.4.8

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

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Step 3.4.4.8.1

Move $x^{2}$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 (x^{2} x^{2}) + x^{2} (- 8 x)$

Step 3.4.4.8.2

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{2 + 2} + x^{2} (- 8 x)$

Step 3.4.4.8.3

Add $2$ and $2$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} + x^{2} (- 8 x)$

Step 3.4.4.9

Rewrite using the commutative property of multiplication.

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} - 8 x^{2} x$

Step 3.4.4.10

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

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Step 3.4.4.10.1

Move $x$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} - 8 (x \cdot x^{2})$

Step 3.4.4.10.2

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

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Step 3.4.4.10.2.1

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} - 8 (x^{1} x^{2})$

Step 3.4.4.10.2.2

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

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} - 8 x^{1 + 2}$

Step 3.4.4.10.3

Add $1$ and $2$ .

$30 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} + 18 x^{7} - 24 x^{6} + 6 x^{4} - 8 x^{3}$

Step 3.5

Add $30 x^{7}$ and $18 x^{7}$ .

$48 x^{7} - 60 x^{6} + 4 x^{4} - 8 x^{3} - 24 x^{6} + 6 x^{4} - 8 x^{3}$

Step 3.6

Subtract $24 x^{6}$ from $- 60 x^{6}$ .

$48 x^{7} - 84 x^{6} + 4 x^{4} - 8 x^{3} + 6 x^{4} - 8 x^{3}$

Step 3.7

Add $4 x^{4}$ and $6 x^{4}$ .

$48 x^{7} - 84 x^{6} + 10 x^{4} - 8 x^{3} - 8 x^{3}$

Step 3.8

Subtract $8 x^{3}$ from $- 8 x^{3}$ .

$48 x^{7} - 84 x^{6} + 10 x^{4} - 16 x^{3}$