Calculus Examples

$f (x) = 8 x + 8 \cos (8 x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $8$ by $1$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [8 \cos (8 x)]$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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Step 1.3.2.1

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

$8 + 8 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [8 x])$

Step 1.3.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

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

Step 1.3.2.3

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Multiply $8$ by $1$ .

$8 + 8 (- \sin (8 x) \cdot 8)$

Step 1.3.6

Multiply $8$ by $- 1$ .

$8 + 8 (- 8 \sin (8 x))$

Step 1.3.7

Multiply $- 8$ by $8$ .

$8 - 64 \sin (8 x)$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $8 - 64 \sin (8 x)$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [- 64 \sin (8 x)]$ .

$f'' (x) = \frac{d}{d x} (8) + \frac{d}{d x} (- 64 \sin (8 x))$

Step 2.1.2

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

$f'' (x) = 0 + \frac{d}{d x} (- 64 \sin (8 x))$

Step 2.2

Evaluate $\frac{d}{d x} [- 64 \sin (8 x)]$ .

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Step 2.2.1

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64 \sin (8 x)$ with respect to $x$ is $- 64 \frac{d}{d x} [\sin (8 x)]$ .

$f'' (x) = 0 - 64 \frac{d}{d x} \sin (8 x)$

Step 2.2.2

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

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Step 2.2.2.1

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

$f'' (x) = 0 - 64 (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (8 x))$

Step 2.2.2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$f'' (x) = 0 - 64 (\cos (u) \frac{d}{d x} (8 x))$

Step 2.2.2.3

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

$f'' (x) = 0 - 64 (\cos (8 x) \frac{d}{d x} (8 x))$

Step 2.2.3

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

$f'' (x) = 0 - 64 (\cos (8 x) (8 \frac{d}{d x} (x)))$

Step 2.2.4

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

$f'' (x) = 0 - 64 (\cos (8 x) (8 \cdot 1))$

Step 2.2.5

Multiply $8$ by $1$ .

$f'' (x) = 0 - 64 (\cos (8 x) \cdot 8)$

Step 2.2.6

Move $8$ to the left of $\cos (8 x)$ .

$f'' (x) = 0 - 64 (8 \cdot \cos (8 x))$

Step 2.2.7

Multiply $8$ by $- 64$ .

$f'' (x) = 0 - 512 \cos (8 x)$

Step 2.3

Subtract $512 \cos (8 x)$ from $0$ .

$f'' (x) = - 512 \cos (8 x)$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$8 - 64 \sin (8 x) = 0$

Step 4

Subtract $8$ from both sides of the equation.

$- 64 \sin (8 x) = - 8$

Step 5

Divide each term in $- 64 \sin (8 x) = - 8$ by $- 64$ and simplify.

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Step 5.1

Divide each term in $- 64 \sin (8 x) = - 8$ by $- 64$ .

$\frac{- 64 \sin (8 x)}{- 64} = \frac{- 8}{- 64}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 64$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 64 \sin (8 x)}{- 64} = \frac{- 8}{- 64}$

Step 5.2.1.2

Divide $\sin (8 x)$ by $1$ .

$\sin (8 x) = \frac{- 8}{- 64}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 8$ and $- 64$ .

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Step 5.3.1.1

Factor $- 8$ out of $- 8$ .

$\sin (8 x) = \frac{- 8 (1)}{- 64}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 8$ out of $- 64$ .

$\sin (8 x) = \frac{- 8 \cdot 1}{- 8 \cdot 8}$

Step 5.3.1.2.2

Cancel the common factor.

$\sin (8 x) = \frac{- 8 \cdot 1}{- 8 \cdot 8}$

Step 5.3.1.2.3

Rewrite the expression.

$\sin (8 x) = \frac{1}{8}$

Step 6

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$8 x = \arcsin (\frac{1}{8})$

Step 7

Simplify the right side.

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Step 7.1

Evaluate $\arcsin (\frac{1}{8})$ .

$8 x = 0.12532783$

Step 8

Divide each term in $8 x = 0.12532783$ by $8$ and simplify.

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Step 8.1

Divide each term in $8 x = 0.12532783$ by $8$ .

$\frac{8 x}{8} = \frac{0.12532783}{8}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $8$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{0.12532783}{8}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.12532783}{8}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $0.12532783$ by $8$ .

$x = 0.01566597$

Step 9

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$8 x = (3.14159265) - 0.12532783$

Step 10

Solve for $x$ .

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Step 10.1

Subtract $0.12532783$ from $3.14159265$ .

$8 x = 3.01626482$

Step 10.2

Divide each term in $8 x = 3.01626482$ by $8$ and simplify.

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Step 10.2.1

Divide each term in $8 x = 3.01626482$ by $8$ .

$\frac{8 x}{8} = \frac{3.01626482}{8}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $8$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{3.01626482}{8}$

Step 10.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3.01626482}{8}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Divide $3.01626482$ by $8$ .

$x = 0.3770331$

Step 11

The solution to the equation $8 x = 0.12532783$ .

$x = 0.01566597, 0.3770331$

Step 12

Evaluate the second derivative at $x = 0.01566597$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 512 \cos (8 (0.01566597))$

Step 13

Multiply $8$ by $0.01566597$ .

$- 512 \cos (0.12532783)$

Step 14

$x = 0.01566597$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 0.01566597$ is a local maximum

Step 15

Find the y-value when $x = 0.01566597$ .

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Step 15.1

Replace the variable $x$ with $0.01566597$ in the expression.

$f (0.01566597) = 8 (0.01566597) + 8 \cos (8 (0.01566597))$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Multiply $8$ by $0.01566597$ .

$f (0.01566597) = 0.12532783 + 8 \cos (8 (0.01566597))$

Step 15.2.1.2

Multiply $8$ by $0.01566597$ .

$f (0.01566597) = 0.12532783 + 8 \cos (0.12532783)$

Step 15.2.2

Add $0.12532783$ and $8 \cos (0.12532783)$ .

$f (0.01566597) = 8.06258176$

Step 15.2.3

The final answer is $8.06258176$ .

$y = 8.06258176$

Step 16

Evaluate the second derivative at $x = 0.3770331$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 512 \cos (8 (0.3770331))$

Step 17

Multiply $8$ by $0.3770331$ .

$- 512 \cos (3.01626482)$

Step 18

$x = 0.3770331$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0.3770331$ is a local minimum

Step 19

Find the y-value when $x = 0.3770331$ .

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Step 19.1

Replace the variable $x$ with $0.3770331$ in the expression.

$f (0.3770331) = 8 (0.3770331) + 8 \cos (8 (0.3770331))$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Multiply $8$ by $0.3770331$ .

$f (0.3770331) = 3.01626482 + 8 \cos (8 (0.3770331))$

Step 19.2.1.2

Multiply $8$ by $0.3770331$ .

$f (0.3770331) = 3.01626482 + 8 \cos (3.01626482)$

Step 19.2.2

Add $3.01626482$ and $8 \cos (3.01626482)$ .

$f (0.3770331) = - 4.92098911$

Step 19.2.3

The final answer is $- 4.92098911$ .

$y = - 4.92098911$

Step 20

These are the local extrema for $f (x) = 8 x + 8 \cos (8 x)$ .

$(0.01566597, 8.06258176)$ is a local maxima

$(0.3770331, - 4.92098911)$ is a local minima

Step 21