Calculus Examples

$f (x) = - 5 \sin (\frac{1}{3} x) - 15$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $- 5 \sin (\frac{1}{3} x) - 15$ with respect to $x$ is $\frac{d}{d x} [- 5 \sin (\frac{1}{3} x)] + \frac{d}{d x} [- 15]$ .

$\frac{d}{d x} [- 5 \sin (\frac{1}{3} x)] + \frac{d}{d x} [- 15]$

Step 1.2

Evaluate $\frac{d}{d x} [- 5 \sin (\frac{1}{3} x)]$ .

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Step 1.2.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{d}{d x} [- 5 \sin (\frac{x}{3})] + \frac{d}{d x} [- 15]$

Step 1.2.2

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \sin (\frac{x}{3})$ with respect to $x$ is $- 5 \frac{d}{d x} [\sin (\frac{x}{3})]$ .

$- 5 \frac{d}{d x} [\sin (\frac{x}{3})] + \frac{d}{d x} [- 15]$

Step 1.2.3

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

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Step 1.2.3.1

To apply the Chain Rule, set $u$ as $\frac{x}{3}$ .

$- 5 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [\frac{x}{3}]) + \frac{d}{d x} [- 15]$

Step 1.2.3.2

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

$- 5 (\cos (u) \frac{d}{d x} [\frac{x}{3}]) + \frac{d}{d x} [- 15]$

Step 1.2.3.3

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$- 5 (\cos (\frac{x}{3}) \frac{d}{d x} [\frac{x}{3}]) + \frac{d}{d x} [- 15]$

Step 1.2.4

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$- 5 (\cos (\frac{x}{3}) (\frac{1}{3} \frac{d}{d x} [x])) + \frac{d}{d x} [- 15]$

Step 1.2.5

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

$- 5 (\cos (\frac{x}{3}) (\frac{1}{3} \cdot 1)) + \frac{d}{d x} [- 15]$

Step 1.2.6

Multiply $\frac{1}{3}$ by $1$ .

$- 5 (\cos (\frac{x}{3}) \frac{1}{3}) + \frac{d}{d x} [- 15]$

Step 1.2.7

Combine $\cos (\frac{x}{3})$ and $\frac{1}{3}$ .

$- 5 \frac{\cos (\frac{x}{3})}{3} + \frac{d}{d x} [- 15]$

Step 1.2.8

Combine $- 5$ and $\frac{\cos (\frac{x}{3})}{3}$ .

$\frac{- 5 \cos (\frac{x}{3})}{3} + \frac{d}{d x} [- 15]$

Step 1.2.9

Move the negative in front of the fraction.

$- \frac{5 \cos (\frac{x}{3})}{3} + \frac{d}{d x} [- 15]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$- \frac{5 \cos (\frac{x}{3})}{3} + 0$

Step 1.3.2

Add $- \frac{5 \cos (\frac{x}{3})}{3}$ and $0$ .

$- \frac{5 \cos (\frac{x}{3})}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- \frac{5}{3}$ is constant with respect to $x$ , the derivative of $- \frac{5 \cos (\frac{x}{3})}{3}$ with respect to $x$ is $- \frac{5}{3} \frac{d}{d x} [\cos (\frac{x}{3})]$ .

$f'' (x) = - \frac{5}{3} \cdot \frac{d}{d x} \cos (\frac{x}{3})$

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{3}$ .

$f'' (x) = - \frac{5}{3} \cdot (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (\frac{x}{3}))$

Step 2.2.2

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

$f'' (x) = - \frac{5}{3} \cdot (- \sin (u) \frac{d}{d x} \frac{x}{3})$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$f'' (x) = - \frac{5}{3} \cdot (- \sin (\frac{x}{3}) \frac{d}{d x} \frac{x}{3})$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{5}{3}) \cdot (\sin (\frac{x}{3}) \frac{d}{d x} (\frac{x}{3}))$

Step 2.3.2

Combine fractions.

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Step 2.3.2.1

Multiply $\frac{5}{3}$ by $1$ .

$f'' (x) = \frac{5}{3} \cdot (\sin (\frac{x}{3}) \frac{d}{d x} (\frac{x}{3}))$

Step 2.3.2.2

Combine $\sin (\frac{x}{3})$ and $\frac{5}{3}$ .

$f'' (x) = \frac{\sin (\frac{x}{3}) \cdot 5}{3} \cdot \frac{d}{d x} (\frac{x}{3})$

Step 2.3.2.3

Move $5$ to the left of $\sin (\frac{x}{3})$ .

$f'' (x) = \frac{5 \cdot \sin (\frac{x}{3})}{3} \cdot \frac{d}{d x} (\frac{x}{3})$

Step 2.3.3

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$f'' (x) = \frac{5 \sin (\frac{x}{3})}{3} \cdot (\frac{1}{3} \cdot \frac{d}{d x} (x))$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $\frac{1}{3}$ by $\frac{5 \sin (\frac{x}{3})}{3}$ .

$f'' (x) = \frac{5 \sin (\frac{x}{3})}{3 \cdot 3} \cdot \frac{d}{d x} (x)$

Step 2.3.4.2

Multiply $3$ by $3$ .

$f'' (x) = \frac{5 \sin (\frac{x}{3})}{9} \cdot \frac{d}{d x} (x)$

Step 2.3.5

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

$f'' (x) = \frac{5 \sin (\frac{x}{3})}{9} \cdot 1$

Step 2.3.6

Multiply $\frac{5 \sin (\frac{x}{3})}{9}$ by $1$ .

$f'' (x) = \frac{5 \sin (\frac{x}{3})}{9}$

Step 3

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

$- \frac{5 \cos (\frac{x}{3})}{3} = 0$

Step 4

Set the numerator equal to zero.

$5 \cos (\frac{x}{3}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $5 \cos (\frac{x}{3}) = 0$ by $5$ and simplify.

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Step 5.1.1

Divide each term in $5 \cos (\frac{x}{3}) = 0$ by $5$ .

$\frac{5 \cos (\frac{x}{3})}{5} = \frac{0}{5}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $5$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{5 \cos (\frac{x}{3})}{5} = \frac{0}{5}$

Step 5.1.2.1.2

Divide $\cos (\frac{x}{3})$ by $1$ .

$\cos (\frac{x}{3}) = \frac{0}{5}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $5$ .

$\cos (\frac{x}{3}) = 0$

Step 5.2

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

$\frac{x}{3} = \arccos (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$\frac{x}{3} = \frac{π}{2}$

Step 5.4

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \frac{π}{2}$

Step 5.5

Simplify both sides of the equation.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Cancel the common factor of $3$ .

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Step 5.5.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \frac{π}{2}$

Step 5.5.1.1.2

Rewrite the expression.

$x = 3 \frac{π}{2}$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Combine $3$ and $\frac{π}{2}$ .

$x = \frac{3 π}{2}$

Step 5.6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$\frac{x}{3} = 2 π - \frac{π}{2}$

Step 5.7

Solve for $x$ .

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Step 5.7.1

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 (2 π - \frac{π}{2})$

Step 5.7.2

Simplify both sides of the equation.

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Step 5.7.2.1

Simplify the left side.

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Step 5.7.2.1.1

Cancel the common factor of $3$ .

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Step 5.7.2.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 (2 π - \frac{π}{2})$

Step 5.7.2.1.1.2

Rewrite the expression.

$x = 3 (2 π - \frac{π}{2})$

Step 5.7.2.2

Simplify the right side.

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Step 5.7.2.2.1

Simplify $3 (2 π - \frac{π}{2})$ .

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Step 5.7.2.2.1.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 3 (2 π \cdot \frac{2}{2} - \frac{π}{2})$

Step 5.7.2.2.1.2

Combine fractions.

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Step 5.7.2.2.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = 3 (\frac{2 π \cdot 2}{2} - \frac{π}{2})$

Step 5.7.2.2.1.2.2

Combine the numerators over the common denominator.

$x = 3 \frac{2 π \cdot 2 - π}{2}$

Step 5.7.2.2.1.3

Simplify the numerator.

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Step 5.7.2.2.1.3.1

Multiply $2$ by $2$ .

$x = 3 \frac{4 π - π}{2}$

Step 5.7.2.2.1.3.2

Subtract $π$ from $4 π$ .

$x = 3 \frac{3 π}{2}$

Step 5.7.2.2.1.4

Multiply $3 \frac{3 π}{2}$ .

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Step 5.7.2.2.1.4.1

Combine $3$ and $\frac{3 π}{2}$ .

$x = \frac{3 (3 π)}{2}$

Step 5.7.2.2.1.4.2

Multiply $3$ by $3$ .

$x = \frac{9 π}{2}$

Step 5.8

The solution to the equation $\frac{x}{3} = \frac{π}{2}$ .

$x = \frac{3 π}{2}, \frac{9 π}{2}$

Step 6

Evaluate the second derivative at $x = \frac{3 π}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5 \sin (\frac{\frac{3 π}{2}}{3})}{9}$

Step 7

Evaluate the second derivative.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.1.2

Cancel the common factor of $3$ .

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Step 7.1.2.1

Factor $3$ out of $3 π$ .

$\frac{5 \sin (\frac{3 (π)}{2} \cdot \frac{1}{3})}{9}$

Step 7.1.2.2

Cancel the common factor.

$\frac{5 \sin (\frac{3 π}{2} \cdot \frac{1}{3})}{9}$

Step 7.1.2.3

Rewrite the expression.

$\frac{5 \sin (\frac{π}{2})}{9}$

Step 7.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{5 \cdot 1}{9}$

Step 7.2

Multiply $5$ by $1$ .

$\frac{5}{9}$

Step 8

$x = \frac{3 π}{2}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{2}$ is a local minimum

Step 9

Find the y-value when $x = \frac{3 π}{2}$ .

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Step 9.1

Replace the variable $x$ with $\frac{3 π}{2}$ in the expression.

$f (\frac{3 π}{2}) = - 5 \sin (\frac{1}{3} \cdot (\frac{3 π}{2})) - 15$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

Cancel the common factor of $3$ .

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Step 9.2.1.1.1

Factor $3$ out of $3 π$ .

$f (\frac{3 π}{2}) = - 5 \sin (\frac{1}{3} \cdot \frac{3 (π)}{2}) - 15$

Step 9.2.1.1.2

Cancel the common factor.

$f (\frac{3 π}{2}) = - 5 \sin (\frac{1}{3} \cdot \frac{3 π}{2}) - 15$

Step 9.2.1.1.3

Rewrite the expression.

$f (\frac{3 π}{2}) = - 5 \sin (\frac{π}{2}) - 15$

Step 9.2.1.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{3 π}{2}) = - 5 \cdot 1 - 15$

Step 9.2.1.3

Multiply $- 5$ by $1$ .

$f (\frac{3 π}{2}) = - 5 - 15$

Step 9.2.2

Subtract $15$ from $- 5$ .

$f (\frac{3 π}{2}) = - 20$

Step 9.2.3

The final answer is $- 20$ .

$y = - 20$

Step 10

Evaluate the second derivative at $x = \frac{9 π}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{5 \sin (\frac{\frac{9 π}{2}}{3})}{9}$

Step 11

Evaluate the second derivative.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 \sin (\frac{9 π}{2} \cdot \frac{1}{3})}{9}$

Step 11.1.2

Cancel the common factor of $3$ .

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Step 11.1.2.1

Factor $3$ out of $9 π$ .

$\frac{5 \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.1.2.2

Cancel the common factor.

$\frac{5 \sin (\frac{3 (3 π)}{2} \cdot \frac{1}{3})}{9}$

Step 11.1.2.3

Rewrite the expression.

$\frac{5 \sin (\frac{3 π}{2})}{9}$

Step 11.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$\frac{5 (- \sin (\frac{π}{2}))}{9}$

Step 11.1.4

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$\frac{5 (- 1 \cdot 1)}{9}$

Step 11.1.5

Multiply $- 1$ by $1$ .

$\frac{5 \cdot - 1}{9}$

Step 11.2

Simplify the expression.

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Step 11.2.1

Multiply $5$ by $- 1$ .

$\frac{- 5}{9}$

Step 11.2.2

Move the negative in front of the fraction.

$- \frac{5}{9}$

Step 12

$x = \frac{9 π}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{9 π}{2}$ is a local maximum

Step 13

Find the y-value when $x = \frac{9 π}{2}$ .

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Step 13.1

Replace the variable $x$ with $\frac{9 π}{2}$ in the expression.

$f (\frac{9 π}{2}) = - 5 \sin (\frac{1}{3} \cdot (\frac{9 π}{2})) - 15$

Step 13.2

Simplify the result.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Cancel the common factor of $3$ .

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Step 13.2.1.1.1

Factor $3$ out of $9 π$ .

$f (\frac{9 π}{2}) = - 5 \sin (\frac{1}{3} \cdot \frac{3 (3 π)}{2}) - 15$

Step 13.2.1.1.2

Cancel the common factor.

$f (\frac{9 π}{2}) = - 5 \sin (\frac{1}{3} \cdot \frac{3 (3 π)}{2}) - 15$

Step 13.2.1.1.3

Rewrite the expression.

$f (\frac{9 π}{2}) = - 5 \sin (\frac{3 π}{2}) - 15$

Step 13.2.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{9 π}{2}) = - 5 (- \sin (\frac{π}{2})) - 15$

Step 13.2.1.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{9 π}{2}) = - 5 (- 1 \cdot 1) - 15$

Step 13.2.1.4

Multiply $- 5 (- 1 \cdot 1)$ .

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Step 13.2.1.4.1

Multiply $- 1$ by $1$ .

$f (\frac{9 π}{2}) = - 5 \cdot - 1 - 15$

Step 13.2.1.4.2

Multiply $- 5$ by $- 1$ .

$f (\frac{9 π}{2}) = 5 - 15$

Step 13.2.2

Subtract $15$ from $5$ .

$f (\frac{9 π}{2}) = - 10$

Step 13.2.3

The final answer is $- 10$ .

$y = - 10$

Step 14

These are the local extrema for $f (x) = - 5 \sin (\frac{1}{3} x) - 15$ .

$(\frac{3 π}{2}, - 20)$ is a local minima

$(\frac{9 π}{2}, - 10)$ is a local maxima

Step 15