Calculus Examples

$V = x (36 - x) (24 - 2 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (V) = \frac{d}{d x} (x (36 - x) (24 - 2 x))$

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 $f (x) = x (36 - x)$ and $g (x) = 24 - 2 x$ .

$x (36 - x) \frac{d}{d x} [24 - 2 x] + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $24 - 2 x$ with respect to $x$ is $\frac{d}{d x} [24] + \frac{d}{d x} [- 2 x]$ .

$x (36 - x) (\frac{d}{d x} [24] + \frac{d}{d x} [- 2 x]) + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.2

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$x (36 - x) (0 + \frac{d}{d x} [- 2 x]) + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$x (36 - x) \frac{d}{d x} [- 2 x] + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.4

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

$x (36 - x) (- 2 \frac{d}{d x} [x]) + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.5

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

$x (36 - x) (- 2 \cdot 1) + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Multiply $- 2$ by $1$ .

$x (36 - x) \cdot - 2 + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

Step 3.2.6.2

Move $- 2$ to the left of $x (36 - x)$ .

$- 2 \cdot (x (36 - x)) + (24 - 2 x) \frac{d}{d x} [x (36 - x)]$

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 $f (x) = x$ and $g (x) = 36 - x$ .

$- 2 x (36 - x) + (24 - 2 x) (x \frac{d}{d x} [36 - x] + (36 - x) \frac{d}{d x} [x])$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $36 - x$ with respect to $x$ is $\frac{d}{d x} [36] + \frac{d}{d x} [- x]$ .

$- 2 x (36 - x) + (24 - 2 x) (x (\frac{d}{d x} [36] + \frac{d}{d x} [- x]) + (36 - x) \frac{d}{d x} [x])$

Step 3.4.2

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$- 2 x (36 - x) + (24 - 2 x) (x (0 + \frac{d}{d x} [- x]) + (36 - x) \frac{d}{d x} [x])$

Step 3.4.3

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

$- 2 x (36 - x) + (24 - 2 x) (x \frac{d}{d x} [- x] + (36 - x) \frac{d}{d x} [x])$

Step 3.4.4

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

$- 2 x (36 - x) + (24 - 2 x) (x (- \frac{d}{d x} [x]) + (36 - x) \frac{d}{d x} [x])$

Step 3.4.5

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

$- 2 x (36 - x) + (24 - 2 x) (x (- 1 \cdot 1) + (36 - x) \frac{d}{d x} [x])$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Multiply $- 1$ by $1$ .

$- 2 x (36 - x) + (24 - 2 x) (x \cdot - 1 + (36 - x) \frac{d}{d x} [x])$

Step 3.4.6.2

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

$- 2 x (36 - x) + (24 - 2 x) (- 1 \cdot x + (36 - x) \frac{d}{d x} [x])$

Step 3.4.6.3

Rewrite $- 1 x$ as $- x$ .

$- 2 x (36 - x) + (24 - 2 x) (- x + (36 - x) \frac{d}{d x} [x])$

Step 3.4.7

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

$- 2 x (36 - x) + (24 - 2 x) (- x + (36 - x) \cdot 1)$

Step 3.4.8

Simplify by adding terms.

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Step 3.4.8.1

Multiply $36 - x$ by $1$ .

$- 2 x (36 - x) + (24 - 2 x) (- x + 36 - x)$

Step 3.4.8.2

Subtract $x$ from $- x$ .

$- 2 x (36 - x) + (24 - 2 x) (- 2 x + 36)$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$- 2 x \cdot 36 - 2 x (- x) + (24 - 2 x) (- 2 x + 36)$

Step 3.5.2

Apply the distributive property.

$- 2 x \cdot 36 - 2 x (- x) + (24 - 2 x) (- 2 x) + (24 - 2 x) \cdot 36$

Step 3.5.3

Apply the distributive property.

$- 2 x \cdot 36 - 2 x (- x) + 24 (- 2 x) - 2 x (- 2 x) + (24 - 2 x) \cdot 36$

Step 3.5.4

Apply the distributive property.

$- 2 x \cdot 36 - 2 x (- x) + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5

Combine terms.

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Step 3.5.5.1

Multiply $36$ by $- 2$ .

$- 72 x - 2 x (- x) + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.2

Multiply $- 1$ by $- 2$ .

$- 72 x + 2 x \cdot x + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.3

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

$- 72 x + 2 (x^{1} x) + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.4

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

$- 72 x + 2 (x^{1} x^{1}) + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.5

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

$- 72 x + 2 x^{1 + 1} + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.6

Add $1$ and $1$ .

$- 72 x + 2 x^{2} + 24 (- 2 x) - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.7

Multiply $- 2$ by $24$ .

$- 72 x + 2 x^{2} - 48 x - 2 x (- 2 x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.8

Multiply $- 2$ by $- 2$ .

$- 72 x + 2 x^{2} - 48 x + 4 x \cdot x + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.9

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

$- 72 x + 2 x^{2} - 48 x + 4 (x^{1} x) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.10

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

$- 72 x + 2 x^{2} - 48 x + 4 (x^{1} x^{1}) + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.11

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

$- 72 x + 2 x^{2} - 48 x + 4 x^{1 + 1} + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.12

Add $1$ and $1$ .

$- 72 x + 2 x^{2} - 48 x + 4 x^{2} + 24 \cdot 36 - 2 x \cdot 36$

Step 3.5.5.13

Multiply $24$ by $36$ .

$- 72 x + 2 x^{2} - 48 x + 4 x^{2} + 864 - 2 x \cdot 36$

Step 3.5.5.14

Multiply $36$ by $- 2$ .

$- 72 x + 2 x^{2} - 48 x + 4 x^{2} + 864 - 72 x$

Step 3.5.5.15

Subtract $72 x$ from $- 48 x$ .

$- 72 x + 2 x^{2} - 120 x + 4 x^{2} + 864$

Step 3.5.5.16

Subtract $120 x$ from $- 72 x$ .

$2 x^{2} - 192 x + 4 x^{2} + 864$

Step 3.5.5.17

Add $2 x^{2}$ and $4 x^{2}$ .

$6 x^{2} - 192 x + 864$

Step 4

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

$V' = 6 x^{2} - 192 x + 864$

Step 5

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

$\frac{d V}{d x} = 6 x^{2} - 192 x + 864$