Calculus Examples

$y = (4 x^{3} + 8) (2 x^{7} - 3)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ((4 x^{3} + 8) (2 x^{7} - 3))$

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 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 $f (x) = 4 x^{3} + 8$ and $g (x) = 2 x^{7} - 3$ .

$(4 x^{3} + 8) \frac{d}{d x} [2 x^{7} - 3] + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $2 x^{7} - 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{7}] + \frac{d}{d x} [- 3]$ .

$(4 x^{3} + 8) (\frac{d}{d x} [2 x^{7}] + \frac{d}{d x} [- 3]) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{7}$ with respect to $x$ is $2 \frac{d}{d x} [x^{7}]$ .

$(4 x^{3} + 8) (2 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- 3]) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.3

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

$(4 x^{3} + 8) (2 (7 x^{6}) + \frac{d}{d x} [- 3]) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.4

Multiply $7$ by $2$ .

$(4 x^{3} + 8) (14 x^{6} + \frac{d}{d x} [- 3]) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.5

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$(4 x^{3} + 8) (14 x^{6} + 0) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Add $14 x^{6}$ and $0$ .

$(4 x^{3} + 8) (14 x^{6}) + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.6.2

Move $14$ to the left of $4 x^{3} + 8$ .

$14 \cdot (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) \frac{d}{d x} [4 x^{3} + 8]$

Step 3.2.7

By the Sum Rule, the derivative of $4 x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [8]$ .

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [8])$

Step 3.2.8

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [8])$

Step 3.2.9

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

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (4 (3 x^{2}) + \frac{d}{d x} [8])$

Step 3.2.10

Multiply $3$ by $4$ .

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (12 x^{2} + \frac{d}{d x} [8])$

Step 3.2.11

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

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (12 x^{2} + 0)$

Step 3.2.12

Simplify the expression.

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Step 3.2.12.1

Add $12 x^{2}$ and $0$ .

$14 (4 x^{3} + 8) x^{6} + (2 x^{7} - 3) (12 x^{2})$

Step 3.2.12.2

Move $12$ to the left of $2 x^{7} - 3$ .

$14 (4 x^{3} + 8) x^{6} + 12 \cdot (2 x^{7} - 3) x^{2}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$(14 (4 x^{3}) + 14 \cdot 8) x^{6} + 12 (2 x^{7} - 3) x^{2}$

Step 3.3.2

Apply the distributive property.

$14 (4 x^{3}) x^{6} + 14 \cdot 8 x^{6} + 12 (2 x^{7} - 3) x^{2}$

Step 3.3.3

Apply the distributive property.

$14 (4 x^{3}) x^{6} + 14 \cdot 8 x^{6} + (12 (2 x^{7}) + 12 \cdot - 3) x^{2}$

Step 3.3.4

Apply the distributive property.

$14 (4 x^{3}) x^{6} + 14 \cdot 8 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5

Combine terms.

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Step 3.3.5.1

Multiply $4$ by $14$ .

$56 x^{3} x^{6} + 14 \cdot 8 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.2

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

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Step 3.3.5.2.1

Move $x^{6}$ .

$56 (x^{6} x^{3}) + 14 \cdot 8 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.2.2

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

$56 x^{6 + 3} + 14 \cdot 8 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.2.3

Add $6$ and $3$ .

$56 x^{9} + 14 \cdot 8 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.3

Multiply $14$ by $8$ .

$56 x^{9} + 112 x^{6} + 12 (2 x^{7}) x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.4

Multiply $2$ by $12$ .

$56 x^{9} + 112 x^{6} + 24 x^{7} x^{2} + 12 \cdot - 3 x^{2}$

Step 3.3.5.5

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

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Step 3.3.5.5.1

Move $x^{2}$ .

$56 x^{9} + 112 x^{6} + 24 (x^{2} x^{7}) + 12 \cdot - 3 x^{2}$

Step 3.3.5.5.2

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

$56 x^{9} + 112 x^{6} + 24 x^{2 + 7} + 12 \cdot - 3 x^{2}$

Step 3.3.5.5.3

Add $2$ and $7$ .

$56 x^{9} + 112 x^{6} + 24 x^{9} + 12 \cdot - 3 x^{2}$

Step 3.3.5.6

Multiply $12$ by $- 3$ .

$56 x^{9} + 112 x^{6} + 24 x^{9} - 36 x^{2}$

Step 3.3.5.7

Add $56 x^{9}$ and $24 x^{9}$ .

$80 x^{9} + 112 x^{6} - 36 x^{2}$

Step 4

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

$y' = 80 x^{9} + 112 x^{6} - 36 x^{2}$

Step 5

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

$\frac{d y}{d x} = 80 x^{9} + 112 x^{6} - 36 x^{2}$