Calculus Examples

$V = (15 - x) (8 - x) x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (V) = \frac{d}{d x} ((15 - x) (8 - x) x)$

Step 2

The derivative of $V$ with respect to $x$ is $V'$ .

$V'$

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 $f (x) = (15 - x) (8 - x)$ and $g (x) = x$ .

$(15 - x) (8 - x) \frac{d}{d x} [x] + x \frac{d}{d x} [(15 - x) (8 - x)]$

Step 3.2

Differentiate using the Power Rule.

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Step 3.2.1

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

$(15 - x) (8 - x) \cdot 1 + x \frac{d}{d x} [(15 - x) (8 - x)]$

Step 3.2.2

Multiply $15 - x$ by $1$ .

$(15 - x) (8 - x) + x \frac{d}{d x} [(15 - x) (8 - x)]$

Step 3.3

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 $f (x) = 15 - x$ and $g (x) = 8 - x$ .

$(15 - x) (8 - x) + x ((15 - x) \frac{d}{d x} [8 - x] + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4

Differentiate.

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Step 3.4.1

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

$(15 - x) (8 - x) + x ((15 - x) (\frac{d}{d x} [8] + \frac{d}{d x} [- x]) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.2

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

$(15 - x) (8 - x) + x ((15 - x) (0 + \frac{d}{d x} [- x]) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$(15 - x) (8 - x) + x ((15 - x) \frac{d}{d x} [- x] + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(15 - x) (8 - x) + x ((15 - x) (- \frac{d}{d x} [x]) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.5

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

$(15 - x) (8 - x) + x ((15 - x) (- 1 \cdot 1) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Multiply $- 1$ by $1$ .

$(15 - x) (8 - x) + x ((15 - x) \cdot - 1 + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.6.2

Move $- 1$ to the left of $15 - x$ .

$(15 - x) (8 - x) + x (- 1 \cdot (15 - x) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.6.3

Rewrite $- 1 (15 - x)$ as $- (15 - x)$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) \frac{d}{d x} [15 - x])$

Step 3.4.7

By the Sum Rule, the derivative of $15 - x$ with respect to $x$ is $\frac{d}{d x} [15] + \frac{d}{d x} [- x]$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) (\frac{d}{d x} [15] + \frac{d}{d x} [- x]))$

Step 3.4.8

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) (0 + \frac{d}{d x} [- x]))$

Step 3.4.9

Add $0$ and $\frac{d}{d x} [- x]$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) \frac{d}{d x} [- x])$

Step 3.4.10

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) (- \frac{d}{d x} [x]))$

Step 3.4.11

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

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) (- 1 \cdot 1))$

Step 3.4.12

Simplify the expression.

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Step 3.4.12.1

Multiply $- 1$ by $1$ .

$(15 - x) (8 - x) + x (- (15 - x) + (8 - x) \cdot - 1)$

Step 3.4.12.2

Move $- 1$ to the left of $8 - x$ .

$(15 - x) (8 - x) + x (- (15 - x) - 1 \cdot (8 - x))$

Step 3.4.12.3

Rewrite $- 1 (8 - x)$ as $- (8 - x)$ .

$(15 - x) (8 - x) + x (- (15 - x) - (8 - x))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$(15 - x) (8 - x) + x (- 1 \cdot 15 - - x - (8 - x))$

Step 3.5.2

Apply the distributive property.

$(15 - x) (8 - x) + x (- 1 \cdot 15 - - x - 1 \cdot 8 - - x)$

Step 3.5.3

Apply the distributive property.

$(15 - x) (8 - x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.4

Apply the distributive property.

$(15 - x) \cdot 8 + (15 - x) (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.5

Apply the distributive property.

$15 \cdot 8 - x \cdot 8 + (15 - x) (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.6

Apply the distributive property.

$15 \cdot 8 - x \cdot 8 + 15 (- x) - x (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7

Combine terms.

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Step 3.5.7.1

Multiply $15$ by $8$ .

$120 - x \cdot 8 + 15 (- x) - x (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.2

Multiply $8$ by $- 1$ .

$120 - 8 x + 15 (- x) - x (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.3

Multiply $- 1$ by $15$ .

$120 - 8 x - 15 x - x (- x) + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.4

Multiply $- 1$ by $- 1$ .

$120 - 8 x - 15 x + 1 x \cdot x + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.5

Multiply $x$ by $1$ .

$120 - 8 x - 15 x + x \cdot x + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.6

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

$120 - 8 x - 15 x + x^{1} x + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.7

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

$120 - 8 x - 15 x + x^{1} x^{1} + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.8

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

$120 - 8 x - 15 x + x^{1 + 1} + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.9

Add $1$ and $1$ .

$120 - 8 x - 15 x + x^{2} + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.10

Subtract $15 x$ from $- 8 x$ .

$120 - 23 x + x^{2} + x (- 1 \cdot 15) + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.11

Multiply $- 1$ by $15$ .

$120 - 23 x + x^{2} + x \cdot - 15 + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.12

Move $- 15$ to the left of $x$ .

$120 - 23 x + x^{2} - 15 \cdot x + x (- - x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.13

Multiply $- 1$ by $- 1$ .

$120 - 23 x + x^{2} - 15 x + x (1 x) + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.14

Multiply $x$ by $1$ .

$120 - 23 x + x^{2} - 15 x + x \cdot x + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.15

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

$120 - 23 x + x^{2} - 15 x + x^{1} x + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.16

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

$120 - 23 x + x^{2} - 15 x + x^{1} x^{1} + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.17

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

$120 - 23 x + x^{2} - 15 x + x^{1 + 1} + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.18

Add $1$ and $1$ .

$120 - 23 x + x^{2} - 15 x + x^{2} + x (- 1 \cdot 8) + x (- - x)$

Step 3.5.7.19

Multiply $- 1$ by $8$ .

$120 - 23 x + x^{2} - 15 x + x^{2} + x \cdot - 8 + x (- - x)$

Step 3.5.7.20

Move $- 8$ to the left of $x$ .

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 \cdot x + x (- - x)$

Step 3.5.7.21

Multiply $- 1$ by $- 1$ .

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x (1 x)$

Step 3.5.7.22

Multiply $x$ by $1$ .

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x \cdot x$

Step 3.5.7.23

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

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x^{1} x$

Step 3.5.7.24

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

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x^{1} x^{1}$

Step 3.5.7.25

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

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x^{1 + 1}$

Step 3.5.7.26

Add $1$ and $1$ .

$120 - 23 x + x^{2} - 15 x + x^{2} - 8 x + x^{2}$

Step 3.5.7.27

Subtract $8 x$ from $- 15 x$ .

$120 - 23 x + x^{2} + x^{2} - 23 x + x^{2}$

Step 3.5.7.28

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

$120 - 23 x + x^{2} + 2 x^{2} - 23 x$

Step 3.5.7.29

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

$120 - 23 x + 3 x^{2} - 23 x$

Step 3.5.7.30

Subtract $23 x$ from $- 23 x$ .

$120 + 3 x^{2} - 46 x$

Step 3.5.8

Reorder terms.

$3 x^{2} - 46 x + 120$

Step 4

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

$V' = 3 x^{2} - 46 x + 120$

Step 5

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

$\frac{d V}{d x} = 3 x^{2} - 46 x + 120$