Calculus Examples

$y = (2 x^{3} + 8) (3 x^{7} - 9)$

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 $f (x) = 2 x^{3} + 8$ and $g (x) = 3 x^{7} - 9$ .

$(2 x^{3} + 8) \frac{d}{d x} [3 x^{7} - 9] + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{7} - 9$ with respect to $x$ is $\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 9]$ .

$(2 x^{3} + 8) (\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 9]) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{7}$ with respect to $x$ is $3 \frac{d}{d x} [x^{7}]$ .

$(2 x^{3} + 8) (3 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- 9]) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$(2 x^{3} + 8) (3 (7 x^{6}) + \frac{d}{d x} [- 9]) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.4

Multiply $7$ by $3$ .

$(2 x^{3} + 8) (21 x^{6} + \frac{d}{d x} [- 9]) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$(2 x^{3} + 8) (21 x^{6} + 0) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $21 x^{6}$ and $0$ .

$(2 x^{3} + 8) (21 x^{6}) + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.6.2

Move $21$ to the left of $2 x^{3} + 8$ .

$21 \cdot (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) \frac{d}{d x} [2 x^{3} + 8]$

Step 2.7

By the Sum Rule, the derivative of $2 x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [8]$ .

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [8])$

Step 2.8

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

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8])$

Step 2.9

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

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (2 (3 x^{2}) + \frac{d}{d x} [8])$

Step 2.10

Multiply $3$ by $2$ .

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (6 x^{2} + \frac{d}{d x} [8])$

Step 2.11

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (6 x^{2} + 0)$

Step 2.12

Simplify the expression.

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Step 2.12.1

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

$21 (2 x^{3} + 8) x^{6} + (3 x^{7} - 9) (6 x^{2})$

Step 2.12.2

Move $6$ to the left of $3 x^{7} - 9$ .

$21 (2 x^{3} + 8) x^{6} + 6 \cdot (3 x^{7} - 9) x^{2}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(21 (2 x^{3}) + 21 \cdot 8) x^{6} + 6 (3 x^{7} - 9) x^{2}$

Step 3.2

Apply the distributive property.

$21 (2 x^{3}) x^{6} + 21 \cdot 8 x^{6} + 6 (3 x^{7} - 9) x^{2}$

Step 3.3

Apply the distributive property.

$21 (2 x^{3}) x^{6} + 21 \cdot 8 x^{6} + (6 (3 x^{7}) + 6 \cdot - 9) x^{2}$

Step 3.4

Apply the distributive property.

$21 (2 x^{3}) x^{6} + 21 \cdot 8 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5

Combine terms.

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Step 3.5.1

Multiply $2$ by $21$ .

$42 x^{3} x^{6} + 21 \cdot 8 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.2

Multiply $x^{3}$ by $x^{6}$ by adding the exponents.

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Step 3.5.2.1

Move $x^{6}$ .

$42 (x^{6} x^{3}) + 21 \cdot 8 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.2.2

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

$42 x^{6 + 3} + 21 \cdot 8 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.2.3

Add $6$ and $3$ .

$42 x^{9} + 21 \cdot 8 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.3

Multiply $21$ by $8$ .

$42 x^{9} + 168 x^{6} + 6 (3 x^{7}) x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.4

Multiply $3$ by $6$ .

$42 x^{9} + 168 x^{6} + 18 x^{7} x^{2} + 6 \cdot - 9 x^{2}$

Step 3.5.5

Multiply $x^{7}$ by $x^{2}$ by adding the exponents.

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Step 3.5.5.1

Move $x^{2}$ .

$42 x^{9} + 168 x^{6} + 18 (x^{2} x^{7}) + 6 \cdot - 9 x^{2}$

Step 3.5.5.2

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

$42 x^{9} + 168 x^{6} + 18 x^{2 + 7} + 6 \cdot - 9 x^{2}$

Step 3.5.5.3

Add $2$ and $7$ .

$42 x^{9} + 168 x^{6} + 18 x^{9} + 6 \cdot - 9 x^{2}$

Step 3.5.6

Multiply $6$ by $- 9$ .

$42 x^{9} + 168 x^{6} + 18 x^{9} - 54 x^{2}$

Step 3.5.7

Add $42 x^{9}$ and $18 x^{9}$ .

$60 x^{9} + 168 x^{6} - 54 x^{2}$