Calculus Examples

$- 2 \cos (\frac{5}{4} x)$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Combine $\frac{5}{4}$ and $x$ .

$\frac{d}{d x} [- 2 \cos (\frac{5 x}{4})]$

Step 1.1.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 \cos (\frac{5 x}{4})$ with respect to $x$ is $- 2 \frac{d}{d x} [\cos (\frac{5 x}{4})]$ .

$- 2 \frac{d}{d x} [\cos (\frac{5 x}{4})]$

Step 1.2

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 $g (x) = \frac{5 x}{4}$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{5 x}{4}$ .

$- 2 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{5 x}{4}])$

Step 1.2.2

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

$- 2 (- \sin (u) \frac{d}{d x} [\frac{5 x}{4}])$

Step 1.2.3

Replace all occurrences of $u$ with $\frac{5 x}{4}$ .

$- 2 (- \sin (\frac{5 x}{4}) \frac{d}{d x} [\frac{5 x}{4}])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $- 2$ .

$2 (\sin (\frac{5 x}{4}) \frac{d}{d x} [\frac{5 x}{4}])$

Step 1.3.2

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5 x}{4}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [x]$ .

$2 \sin (\frac{5 x}{4}) (\frac{5}{4} \frac{d}{d x} [x])$

Step 1.3.3

Simplify terms.

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Step 1.3.3.1

Combine $\frac{5}{4}$ and $2$ .

$\frac{5 \cdot 2}{4} \sin (\frac{5 x}{4}) \frac{d}{d x} [x]$

Step 1.3.3.2

Multiply $5$ by $2$ .

$\frac{10}{4} \sin (\frac{5 x}{4}) \frac{d}{d x} [x]$

Step 1.3.3.3

Combine $\frac{10}{4}$ and $\sin (\frac{5 x}{4})$ .

$\frac{10 \sin (\frac{5 x}{4})}{4} \frac{d}{d x} [x]$

Step 1.3.3.4

Cancel the common factor of $10$ and $4$ .

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Step 1.3.3.4.1

Factor $2$ out of $10 \sin (\frac{5 x}{4})$ .

$\frac{2 (5 \sin (\frac{5 x}{4}))}{4} \frac{d}{d x} [x]$

Step 1.3.3.4.2

Cancel the common factors.

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Step 1.3.3.4.2.1

Factor $2$ out of $4$ .

$\frac{2 (5 \sin (\frac{5 x}{4}))}{2 (2)} \frac{d}{d x} [x]$

Step 1.3.3.4.2.2

Cancel the common factor.

$\frac{2 (5 \sin (\frac{5 x}{4}))}{2 \cdot 2} \frac{d}{d x} [x]$

Step 1.3.3.4.2.3

Rewrite the expression.

$\frac{5 \sin (\frac{5 x}{4})}{2} \frac{d}{d x} [x]$

Step 1.3.4

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

$\frac{5 \sin (\frac{5 x}{4})}{2} \cdot 1$

Step 1.3.5

Multiply $\frac{5 \sin (\frac{5 x}{4})}{2}$ by $1$ .

$\frac{5 \sin (\frac{5 x}{4})}{2}$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 \sin (\frac{5 x}{4})}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [\sin (\frac{5 x}{4})]$ .

$f'' (x) = \frac{5}{2} \cdot \frac{d}{d x} (\sin (\frac{5 x}{4}))$

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 $f (x) = \sin (x)$ and $g (x) = \frac{5 x}{4}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{5 x}{4}$ .

$f'' (x) = \frac{5}{2} \cdot (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{5 x}{4}))$

Step 2.2.2

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

$f'' (x) = \frac{5}{2} \cdot (\cos (u) \frac{d}{d x} (\frac{5 x}{4}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{5 x}{4}$ .

$f'' (x) = \frac{5}{2} \cdot (\cos (\frac{5 x}{4}) \frac{d}{d x} (\frac{5 x}{4}))$

Step 2.3

Differentiate.

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Step 2.3.1

Combine $\cos (\frac{5 x}{4})$ and $\frac{5}{2}$ .

$f'' (x) = \frac{\cos (\frac{5 x}{4}) \cdot 5}{2} \cdot \frac{d}{d x} (\frac{5 x}{4})$

Step 2.3.2

Move $5$ to the left of $\cos (\frac{5 x}{4})$ .

$f'' (x) = \frac{5 \cdot \cos (\frac{5 x}{4})}{2} \cdot \frac{d}{d x} (\frac{5 x}{4})$

Step 2.3.3

Since $\frac{5}{4}$ is constant with respect to $x$ , the derivative of $\frac{5 x}{4}$ with respect to $x$ is $\frac{5}{4} \frac{d}{d x} [x]$ .

$f'' (x) = \frac{5 \cos (\frac{5 x}{4})}{2} \cdot (\frac{5}{4} \cdot \frac{d}{d x} (x))$

Step 2.3.4

Combine fractions.

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Step 2.3.4.1

Multiply $\frac{5}{4}$ by $\frac{5 \cos (\frac{5 x}{4})}{2}$ .

$f'' (x) = \frac{5 (5 \cos (\frac{5 x}{4}))}{4 \cdot 2} \cdot \frac{d}{d x} (x)$

Step 2.3.4.2

Multiply.

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Step 2.3.4.2.1

Multiply $5$ by $5$ .

$f'' (x) = \frac{25 \cos (\frac{5 x}{4})}{4 \cdot 2} \cdot \frac{d}{d x} (x)$

Step 2.3.4.2.2

Multiply $4$ by $2$ .

$f'' (x) = \frac{25 \cos (\frac{5 x}{4})}{8} \cdot \frac{d}{d x} (x)$

Step 2.3.5

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

$f'' (x) = \frac{25 \cos (\frac{5 x}{4})}{8} \cdot 1$

Step 2.3.6

Multiply $\frac{25 \cos (\frac{5 x}{4})}{8}$ by $1$ .

$f'' (x) = \frac{25 \cos (\frac{5 x}{4})}{8}$

Step 3

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

$\frac{5 \sin (\frac{5 x}{4})}{2} = 0$

Step 4

Set the numerator equal to zero.

$5 \sin (\frac{5 x}{4}) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $5 \sin (\frac{5 x}{4}) = 0$ by $5$ and simplify.

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Step 5.1.1

Divide each term in $5 \sin (\frac{5 x}{4}) = 0$ by $5$ .

$\frac{5 \sin (\frac{5 x}{4})}{5} = \frac{0}{5}$

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.

$\frac{5 \sin (\frac{5 x}{4})}{5} = \frac{0}{5}$

Step 5.1.2.1.2

Divide $\sin (\frac{5 x}{4})$ by $1$ .

$\sin (\frac{5 x}{4}) = \frac{0}{5}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $5$ .

$\sin (\frac{5 x}{4}) = 0$

Step 5.2

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

$\frac{5 x}{4} = \arcsin (0)$

Step 5.3

Simplify the right side.

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Step 5.3.1

The exact value of $\arcsin (0)$ is $0$ .

$\frac{5 x}{4} = 0$

Step 5.4

Set the numerator equal to zero.

$5 x = 0$

Step 5.5

Divide each term in $5 x = 0$ by $5$ and simplify.

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Step 5.5.1

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $5$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

Step 5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{5}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Divide $0$ by $5$ .

$x = 0$

Step 5.6

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.

$\frac{5 x}{4} = π - 0$

Step 5.7

Solve for $x$ .

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Step 5.7.1

Multiply both sides of the equation by $\frac{4}{5}$ .

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (π - 0)$

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

Simplify $\frac{4}{5} \cdot \frac{5 x}{4}$ .

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Step 5.7.2.1.1.1

Cancel the common factor of $4$ .

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Step 5.7.2.1.1.1.1

Cancel the common factor.

$\frac{4}{5} \cdot \frac{5 x}{4} = \frac{4}{5} (π - 0)$

Step 5.7.2.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 x) = \frac{4}{5} (π - 0)$

Step 5.7.2.1.1.2

Cancel the common factor of $5$ .

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Step 5.7.2.1.1.2.1

Factor $5$ out of $5 x$ .

$\frac{1}{5} (5 (x)) = \frac{4}{5} (π - 0)$

Step 5.7.2.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 x) = \frac{4}{5} (π - 0)$

Step 5.7.2.1.1.2.3

Rewrite the expression.

$x = \frac{4}{5} (π - 0)$

Step 5.7.2.2

Simplify the right side.

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Step 5.7.2.2.1

Simplify $\frac{4}{5} (π - 0)$ .

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Step 5.7.2.2.1.1

Subtract $0$ from $π$ .

$x = \frac{4}{5} π$

Step 5.7.2.2.1.2

Combine $\frac{4}{5}$ and $π$ .

$x = \frac{4 π}{5}$

Step 5.8

The solution to the equation $\frac{5 x}{4} = 0$ .

$x = 0, \frac{4 π}{5}$

Step 6

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{25 \cos (\frac{5 (0)}{4})}{8}$

Step 7

Evaluate the second derivative.

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Step 7.1

Cancel the common factor of $0$ and $4$ .

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Step 7.1.1

Factor $4$ out of $5 (0)$ .

$\frac{25 \cos (\frac{4 (5 \cdot (0))}{4})}{8}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $4$ out of $4$ .

$\frac{25 \cos (\frac{4 (5 \cdot (0))}{4 (1)})}{8}$

Step 7.1.2.2

Cancel the common factor.

$\frac{25 \cos (\frac{4 (5 \cdot (0))}{4 \cdot 1})}{8}$

Step 7.1.2.3

Rewrite the expression.

$\frac{25 \cos (\frac{5 \cdot (0)}{1})}{8}$

Step 7.1.2.4

Divide $5 \cdot (0)$ by $1$ .

$\frac{25 \cos (5 \cdot (0))}{8}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Multiply $5$ by $0$ .

$\frac{25 \cos (0)}{8}$

Step 7.2.2

The exact value of $\cos (0)$ is $1$ .

$\frac{25 \cdot 1}{8}$

Step 7.3

Multiply $25$ by $1$ .

$\frac{25}{8}$

Step 8

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 9

Find the y-value when $x = 0$ .

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Step 9.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 2 \cos (\frac{5}{4} \cdot (0))$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $\frac{5}{4}$ by $0$ .

$f (0) = - 2 \cos (0)$

Step 9.2.2

The exact value of $\cos (0)$ is $1$ .

$f (0) = - 2 \cdot 1$

Step 9.2.3

Multiply $- 2$ by $1$ .

$f (0) = - 2$

Step 9.2.4

The final answer is $- 2$ .

$y = - 2$

Step 10

Evaluate the second derivative at $x = \frac{4 π}{5}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{25 \cos (\frac{5 (\frac{4 π}{5})}{4})}{8}$

Step 11

Evaluate the second derivative.

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Step 11.1

Combine $5$ and $\frac{4 π}{5}$ .

$\frac{25 \cos (\frac{\frac{5 (4 π)}{5}}{4})}{8}$

Step 11.2

Multiply $5$ by $4$ .

$\frac{25 \cos (\frac{\frac{20 π}{5}}{4})}{8}$

Step 11.3

Reduce the expression by cancelling the common factors.

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Step 11.3.1

Reduce the expression $\frac{20 π}{5}$ by cancelling the common factors.

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Step 11.3.1.1

Factor $5$ out of $20 π$ .

$\frac{25 \cos (\frac{\frac{5 (4 π)}{5}}{4})}{8}$

Step 11.3.1.2

Factor $5$ out of $5$ .

$\frac{25 \cos (\frac{\frac{5 (4 π)}{5 (1)}}{4})}{8}$

Step 11.3.1.3

Cancel the common factor.

$\frac{25 \cos (\frac{\frac{5 (4 π)}{5 \cdot 1}}{4})}{8}$

Step 11.3.1.4

Rewrite the expression.

$\frac{25 \cos (\frac{\frac{4 π}{1}}{4})}{8}$

Step 11.3.2

Divide $4 π$ by $1$ .

$\frac{25 \cos (\frac{4 π}{4})}{8}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Cancel the common factor of $4$ .

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Step 11.4.1.1

Cancel the common factor.

$\frac{25 \cos (\frac{4 π}{4})}{8}$

Step 11.4.1.2

Divide $π$ by $1$ .

$\frac{25 \cos (π)}{8}$

Step 11.4.2

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

$\frac{25 (- \cos (0))}{8}$

Step 11.4.3

The exact value of $\cos (0)$ is $1$ .

$\frac{25 (- 1 \cdot 1)}{8}$

Step 11.4.4

Multiply $- 1$ by $1$ .

$\frac{25 \cdot - 1}{8}$

Step 11.5

Simplify the expression.

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Step 11.5.1

Multiply $25$ by $- 1$ .

$\frac{- 25}{8}$

Step 11.5.2

Move the negative in front of the fraction.

$- \frac{25}{8}$

Step 12

$x = \frac{4 π}{5}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{4 π}{5}$ is a local maximum

Step 13

Find the y-value when $x = \frac{4 π}{5}$ .

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Step 13.1

Replace the variable $x$ with $\frac{4 π}{5}$ in the expression.

$f (\frac{4 π}{5}) = - 2 \cos (\frac{5}{4} \cdot (\frac{4 π}{5}))$

Step 13.2

Simplify the result.

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Step 13.2.1

Cancel the common factor of $5$ .

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Step 13.2.1.1

Cancel the common factor.

$f (\frac{4 π}{5}) = - 2 \cos (\frac{5}{4} \cdot \frac{4 π}{5})$

Step 13.2.1.2

Rewrite the expression.

$f (\frac{4 π}{5}) = - 2 \cos (\frac{1}{4} \cdot (4 π))$

Step 13.2.2

Cancel the common factor of $4$ .

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Step 13.2.2.1

Factor $4$ out of $4 π$ .

$f (\frac{4 π}{5}) = - 2 \cos (\frac{1}{4} \cdot (4 (π)))$

Step 13.2.2.2

Cancel the common factor.

$f (\frac{4 π}{5}) = - 2 \cos (\frac{1}{4} \cdot (4 π))$

Step 13.2.2.3

Rewrite the expression.

$f (\frac{4 π}{5}) = - 2 \cos (π)$

Step 13.2.3

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

$f (\frac{4 π}{5}) = - 2 (- \cos (0))$

Step 13.2.4

The exact value of $\cos (0)$ is $1$ .

$f (\frac{4 π}{5}) = - 2 (- 1 \cdot 1)$

Step 13.2.5

Multiply $- 2 (- 1 \cdot 1)$ .

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Step 13.2.5.1

Multiply $- 1$ by $1$ .

$f (\frac{4 π}{5}) = - 2 \cdot - 1$

Step 13.2.5.2

Multiply $- 2$ by $- 1$ .

$f (\frac{4 π}{5}) = 2$

Step 13.2.6

The final answer is $2$ .

$y = 2$

Step 14

These are the local extrema for $f (x) = - 2 \cos (\frac{5}{4} x)$ .

$(0, - 2)$ is a local minima

$(\frac{4 π}{5}, 2)$ is a local maxima

Step 15