Calculus Examples

$f (x) = 3 x {(8 - x)}^{3}$

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

Since $3$ is constant with respect to $x$ , the derivative of $3 x {(8 - x)}^{3}$ with respect to $x$ is $3 \frac{d}{d x} [x {(8 - x)}^{3}]$ .

$f' (x) = 3 \frac{d}{d x} (x {(8 - x)}^{3})$

Step 1.1.2

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 $g (x) = {(8 - x)}^{3}$ .

$f' (x) = 3 (x \frac{d}{d x} ({(8 - x)}^{3}) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 8 - x$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $8 - x$ .

$f' (x) = 3 (x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (8 - x)) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.2

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

$f' (x) = 3 (x (3 u^{2} \frac{d}{d x} (8 - x)) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.3.3

Replace all occurrences of $u$ with $8 - x$ .

$f' (x) = 3 (x (3 {(8 - x)}^{2} \frac{d}{d x} (8 - x)) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4

Differentiate.

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Step 1.1.4.1

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

$f' (x) = 3 (x (3 {(8 - x)}^{2} (\frac{d}{d x} (8) + \frac{d}{d x} (- x))) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.2

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

$f' (x) = 3 (x (3 {(8 - x)}^{2} (0 + \frac{d}{d x} (- x))) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.3

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

$f' (x) = 3 (x (3 {(8 - x)}^{2} \frac{d}{d x} (- x)) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.4

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

$f' (x) = 3 (x (3 {(8 - x)}^{2} (- \frac{d}{d x} x)) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.5

Multiply $- 1$ by $3$ .

$f' (x) = 3 (x (- 3 {(8 - x)}^{2} \frac{d}{d x} x) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.6

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

$f' (x) = 3 (x (- 3 {(8 - x)}^{2} \cdot 1) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.7

Multiply $- 3$ by $1$ .

$f' (x) = 3 (x (- 3 {(8 - x)}^{2}) + {(8 - x)}^{3} \frac{d}{d x} (x))$

Step 1.1.4.8

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

$f' (x) = 3 (x (- 3 {(8 - x)}^{2}) + {(8 - x)}^{3} \cdot 1)$

Step 1.1.4.9

Multiply ${(8 - x)}^{3}$ by $1$ .

$f' (x) = 3 (x (- 3 {(8 - x)}^{2}) + {(8 - x)}^{3})$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

$f' (x) = 3 (x (- 3 {(8 - x)}^{2})) + 3 {(8 - x)}^{3}$

Step 1.1.5.2

Multiply $- 3$ by $3$ .

$f' (x) = - 9 (x ({(8 - x)}^{2})) + 3 {(8 - x)}^{3}$

Step 1.1.5.3

Factor $3 {(8 - x)}^{2}$ out of $- 9 x {(8 - x)}^{2} + 3 {(8 - x)}^{3}$ .

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Step 1.1.5.3.1

Factor $3 {(8 - x)}^{2}$ out of $- 9 x {(8 - x)}^{2}$ .

$f' (x) = 3 {(8 - x)}^{2} (- 3 x) + 3 {(8 - x)}^{3}$

Step 1.1.5.3.2

Factor $3 {(8 - x)}^{2}$ out of $3 {(8 - x)}^{3}$ .

$f' (x) = 3 {(8 - x)}^{2} (- 3 x) + 3 {(8 - x)}^{2} (8 - x)$

Step 1.1.5.3.3

Factor $3 {(8 - x)}^{2}$ out of $3 {(8 - x)}^{2} (- 3 x) + 3 {(8 - x)}^{2} (8 - x)$ .

$f' (x) = 3 {(8 - x)}^{2} (- 3 x + 8 - x)$

Step 1.1.5.4

Subtract $x$ from $- 3 x$ .

$f' (x) = 3 {(8 - x)}^{2} (- 4 x + 8)$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 {(8 - x)}^{2} (- 4 x + 8)$ .

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

Step 2

Set the first derivative equal to $0$ then solve the equation $3 {(8 - x)}^{2} (- 4 x + 8) = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 {(8 - x)}^{2} (- 4 x + 8) = 0$

Step 2.2

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

${(8 - x)}^{2} = 0$

$- 4 x + 8 = 0$

Step 2.3

Set ${(8 - x)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(8 - x)}^{2}$ equal to $0$ .

${(8 - x)}^{2} = 0$

Step 2.3.2

Solve ${(8 - x)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $8 - x$ equal to $0$ .

$8 - x = 0$

Step 2.3.2.2

Solve for $x$ .

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Step 2.3.2.2.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 2.3.2.2.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 2.3.2.2.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 2.3.2.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 2.3.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 2.3.2.2.2.3

Simplify the right side.

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Step 2.3.2.2.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 2.4

Set $- 4 x + 8$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $- 4 x + 8$ equal to $0$ .

$- 4 x + 8 = 0$

Step 2.4.2

Solve $- 4 x + 8 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $8$ from both sides of the equation.

$- 4 x = - 8$

Step 2.4.2.2

Divide each term in $- 4 x = - 8$ by $- 4$ and simplify.

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Step 2.4.2.2.1

Divide each term in $- 4 x = - 8$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 8}{- 4}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 4}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Divide $- 8$ by $- 4$ .

$x = 2$

Step 2.5

The final solution is all the values that make $3 {(8 - x)}^{2} (- 4 x + 8) = 0$ true.

$x = 8, 2$

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 $3 x {(8 - x)}^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 8$ .

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Step 4.1.1

Substitute $8$ for $x$ .

$3 (8) {(8 - (8))}^{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $3$ by $8$ .

$24 {(8 - (8))}^{3}$

Step 4.1.2.2

Multiply $- 1$ by $8$ .

$24 {(8 - 8)}^{3}$

Step 4.1.2.3

Subtract $8$ from $8$ .

$24 \cdot 0^{3}$

Step 4.1.2.4

Raising $0$ to any positive power yields $0$ .

$24 \cdot 0$

Step 4.1.2.5

Multiply $24$ by $0$ .

$0$

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

$3 (2) {(8 - (2))}^{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Multiply $3$ by $2$ .

$6 {(8 - (2))}^{3}$

Step 4.2.2.2

Multiply $- 1$ by $2$ .

$6 {(8 - 2)}^{3}$

Step 4.2.2.3

Subtract $2$ from $8$ .

$6 \cdot 6^{3}$

Step 4.2.2.4

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

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Step 4.2.2.4.1

Multiply $6$ by $6^{3}$ .

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Step 4.2.2.4.1.1

Raise $6$ to the power of $1$ .

$6^{1} \cdot 6^{3}$

Step 4.2.2.4.1.2

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

$6^{1 + 3}$

Step 4.2.2.4.2

Add $1$ and $3$ .

$6^{4}$

Step 4.2.2.5

Raise $6$ to the power of $4$ .

$1296$

Step 4.3

List all of the points.

$(8, 0), (2, 1296)$

Step 5