Calculus Examples

$(24 - 2 x) (24 - 2 x) x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Raise $24 - 2 x$ to the power of $1$ .

$\frac{d}{d x} [{(24 - 2 x)}^{1} (24 - 2 x) x]$

Step 1.1.2

Raise $24 - 2 x$ to the power of $1$ .

$\frac{d}{d x} [{(24 - 2 x)}^{1} {(24 - 2 x)}^{1} x]$

Step 1.1.3

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

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

Step 1.1.4

Add $1$ and $1$ .

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

Step 1.1.5

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 $f (x) = {(24 - 2 x)}^{2}$ and $g (x) = x$ .

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

Step 1.1.6

Differentiate using the Power Rule.

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Step 1.1.6.1

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

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

Step 1.1.6.2

Multiply ${(24 - 2 x)}^{2}$ by $1$ .

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

Step 1.1.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 24 - 2 x$ .

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Step 1.1.7.1

To apply the Chain Rule, set $u$ as $24 - 2 x$ .

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

Step 1.1.7.2

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

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

Step 1.1.7.3

Replace all occurrences of $u$ with $24 - 2 x$ .

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

Step 1.1.8

Differentiate.

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Step 1.1.8.1

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

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

Step 1.1.8.2

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

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

Step 1.1.8.3

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

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

Step 1.1.8.4

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

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

Step 1.1.8.5

Multiply $- 2$ by $2$ .

${(24 - 2 x)}^{2} + x (- 4 (24 - 2 x) \frac{d}{d x} [x])$

Step 1.1.8.6

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

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

Step 1.1.8.7

Multiply $- 4$ by $1$ .

${(24 - 2 x)}^{2} + x (- 4 (24 - 2 x))$

Step 1.1.9

Simplify.

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Step 1.1.9.1

Apply the distributive property.

${(24 - 2 x)}^{2} + x (- 4 \cdot 24 - 4 (- 2 x))$

Step 1.1.9.2

Apply the distributive property.

${(24 - 2 x)}^{2} + x (- 4 \cdot 24) + x (- 4 (- 2 x))$

Step 1.1.9.3

Combine terms.

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Step 1.1.9.3.1

Multiply $- 4$ by $24$ .

${(24 - 2 x)}^{2} + x \cdot - 96 + x (- 4 (- 2 x))$

Step 1.1.9.3.2

Move $- 96$ to the left of $x$ .

${(24 - 2 x)}^{2} - 96 \cdot x + x (- 4 (- 2 x))$

Step 1.1.9.3.3

Multiply $- 2$ by $- 4$ .

${(24 - 2 x)}^{2} - 96 x + x (8 x)$

Step 1.1.9.3.4

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

${(24 - 2 x)}^{2} - 96 x + 8 (x^{1} x)$

Step 1.1.9.3.5

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

${(24 - 2 x)}^{2} - 96 x + 8 (x^{1} x^{1})$

Step 1.1.9.3.6

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

${(24 - 2 x)}^{2} - 96 x + 8 x^{1 + 1}$

Step 1.1.9.3.7

Add $1$ and $1$ .

${(24 - 2 x)}^{2} - 96 x + 8 x^{2}$

Step 1.1.9.4

Reorder terms.

$8 x^{2} + {(24 - 2 x)}^{2} - 96 x$

Step 1.1.9.5

Simplify each term.

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Step 1.1.9.5.1

Rewrite ${(24 - 2 x)}^{2}$ as $(24 - 2 x) (24 - 2 x)$ .

$8 x^{2} + (24 - 2 x) (24 - 2 x) - 96 x$

Step 1.1.9.5.2

Expand $(24 - 2 x) (24 - 2 x)$ using the FOIL Method.

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Step 1.1.9.5.2.1

Apply the distributive property.

$8 x^{2} + 24 (24 - 2 x) - 2 x (24 - 2 x) - 96 x$

Step 1.1.9.5.2.2

Apply the distributive property.

$8 x^{2} + 24 \cdot 24 + 24 (- 2 x) - 2 x (24 - 2 x) - 96 x$

Step 1.1.9.5.2.3

Apply the distributive property.

$8 x^{2} + 24 \cdot 24 + 24 (- 2 x) - 2 x \cdot 24 - 2 x (- 2 x) - 96 x$

Step 1.1.9.5.3

Simplify and combine like terms.

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Step 1.1.9.5.3.1

Simplify each term.

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Step 1.1.9.5.3.1.1

Multiply $24$ by $24$ .

$8 x^{2} + 576 + 24 (- 2 x) - 2 x \cdot 24 - 2 x (- 2 x) - 96 x$

Step 1.1.9.5.3.1.2

Multiply $- 2$ by $24$ .

$8 x^{2} + 576 - 48 x - 2 x \cdot 24 - 2 x (- 2 x) - 96 x$

Step 1.1.9.5.3.1.3

Multiply $24$ by $- 2$ .

$8 x^{2} + 576 - 48 x - 48 x - 2 x (- 2 x) - 96 x$

Step 1.1.9.5.3.1.4

Rewrite using the commutative property of multiplication.

$8 x^{2} + 576 - 48 x - 48 x - 2 \cdot - 2 x \cdot x - 96 x$

Step 1.1.9.5.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.9.5.3.1.5.1

Move $x$ .

$8 x^{2} + 576 - 48 x - 48 x - 2 \cdot - 2 (x \cdot x) - 96 x$

Step 1.1.9.5.3.1.5.2

Multiply $x$ by $x$ .

$8 x^{2} + 576 - 48 x - 48 x - 2 \cdot - 2 x^{2} - 96 x$

Step 1.1.9.5.3.1.6

Multiply $- 2$ by $- 2$ .

$8 x^{2} + 576 - 48 x - 48 x + 4 x^{2} - 96 x$

Step 1.1.9.5.3.2

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

$8 x^{2} + 576 - 96 x + 4 x^{2} - 96 x$

Step 1.1.9.6

Add $8 x^{2}$ and $4 x^{2}$ .

$12 x^{2} + 576 - 96 x - 96 x$

Step 1.1.9.7

Subtract $96 x$ from $- 96 x$ .

$f' (x) = 12 x^{2} + 576 - 192 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} + 576 - 192 x$ .

$12 x^{2} + 576 - 192 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} + 576 - 192 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$12 x^{2} + 576 - 192 x = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $12$ out of $12 x^{2} + 576 - 192 x$ .

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Step 2.2.1.1

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) + 576 - 192 x = 0$

Step 2.2.1.2

Factor $12$ out of $576$ .

$12 x^{2} + 12 \cdot 48 - 192 x = 0$

Step 2.2.1.3

Factor $12$ out of $- 192 x$ .

$12 x^{2} + 12 \cdot 48 + 12 (- 16 x) = 0$

Step 2.2.1.4

Factor $12$ out of $12 x^{2} + 12 \cdot 48$ .

$12 (x^{2} + 48) + 12 (- 16 x) = 0$

Step 2.2.1.5

Factor $12$ out of $12 (x^{2} + 48) + 12 (- 16 x)$ .

$12 (x^{2} + 48 - 16 x) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} + 48 - 16 x$ using the AC method.

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Step 2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $48$ and whose sum is $- 16$ .

$- 12, - 4$

Step 2.2.2.1.2

Write the factored form using these integers.

$12 ((x - 12) (x - 4)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$12 (x - 12) (x - 4) = 0$

Step 2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 12 = 0$

$x - 4 = 0$

Step 2.4

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 2.4.2

Add $12$ to both sides of the equation.

$x = 12$

Step 2.5

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.5.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.6

The final solution is all the values that make $12 (x - 12) (x - 4) = 0$ true.

$x = 12, 4$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 4

Evaluate $(24 - 2 x) (24 - 2 x) x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 12$ .

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Step 4.1.1

Substitute $12$ for $x$ .

$(24 - 2 \cdot 12) (24 - 2 \cdot 12) (12)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 2$ by $12$ .

$(24 - 24) (24 - 2 \cdot 12) \cdot 12$

Step 4.1.2.2

Subtract $24$ from $24$ .

$0 (24 - 2 \cdot 12) \cdot 12$

Step 4.1.2.3

Multiply $- 2$ by $12$ .

$0 (24 - 24) \cdot 12$

Step 4.1.2.4

Subtract $24$ from $24$ .

$0 \cdot 0 \cdot 12$

Step 4.1.2.5

Multiply $0 \cdot 0 \cdot 12$ .

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Step 4.1.2.5.1

Multiply $0$ by $0$ .

$0 \cdot 12$

Step 4.1.2.5.2

Multiply $0$ by $12$ .

$0$

Step 4.2

Evaluate at $x = 4$ .

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Step 4.2.1

Substitute $4$ for $x$ .

$(24 - 2 \cdot 4) (24 - 2 \cdot 4) (4)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $- 2$ by $4$ .

$(24 - 8) (24 - 2 \cdot 4) \cdot 4$

Step 4.2.2.2

Subtract $8$ from $24$ .

$16 (24 - 2 \cdot 4) \cdot 4$

Step 4.2.2.3

Multiply $- 2$ by $4$ .

$16 (24 - 8) \cdot 4$

Step 4.2.2.4

Subtract $8$ from $24$ .

$16 \cdot 16 \cdot 4$

Step 4.2.2.5

Multiply $16 \cdot 16 \cdot 4$ .

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Step 4.2.2.5.1

Multiply $16$ by $16$ .

$256 \cdot 4$

Step 4.2.2.5.2

Multiply $256$ by $4$ .

$1024$

Step 4.3

List all of the points.

$(12, 0), (4, 1024)$

Step 5