Calculus Examples

$y = 2 - \frac{\sin (5 x)}{3}$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $2 - \frac{\sin (5 x)}{3}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- \frac{\sin (5 x)}{3}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [- \frac{\sin (5 x)}{3}]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{\sin (5 x)}{3}]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

Since $- \frac{1}{3}$ is constant with respect to $x$ , the derivative of $- \frac{\sin (5 x)}{3}$ with respect to $x$ is $- \frac{1}{3} \frac{d}{d x} [\sin (5 x)]$ .

$0 - \frac{1}{3} \frac{d}{d x} [\sin (5 x)]$

Step 1.2.2

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 $g (x) = 5 x$ .

Tap for more steps...

Step 1.2.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$0 - \frac{1}{3} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x])$

Step 1.2.2.2

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

$0 - \frac{1}{3} (\cos (u) \frac{d}{d x} [5 x])$

Step 1.2.2.3

Replace all occurrences of $u$ with $5 x$ .

$0 - \frac{1}{3} (\cos (5 x) \frac{d}{d x} [5 x])$

Step 1.2.3

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

$0 - \frac{1}{3} (\cos (5 x) (5 \frac{d}{d x} [x]))$

Step 1.2.4

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

$0 - \frac{1}{3} (\cos (5 x) (5 \cdot 1))$

Step 1.2.5

Multiply $5$ by $1$ .

$0 - \frac{1}{3} (\cos (5 x) \cdot 5)$

Step 1.2.6

Move $5$ to the left of $\cos (5 x)$ .

$0 - \frac{1}{3} (5 \cdot \cos (5 x))$

Step 1.2.7

Multiply $5$ by $- 1$ .

$0 - 5 (\frac{1}{3}) \cos (5 x)$

Step 1.2.8

Combine $- 5$ and $\frac{1}{3}$ .

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

Step 1.2.9

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

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

Step 1.2.10

Move the negative in front of the fraction.

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

Step 1.3

Subtract $\frac{5 \cos (5 x)}{3}$ from $0$ .

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

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

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

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 $g (x) = 5 x$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $5 x$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $5 x$ .

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

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.2

Combine fractions.

Tap for more steps...

Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Move $5$ to the left of $\sin (5 x)$ .

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

Step 2.3.3

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

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

Step 2.3.4

Combine fractions.

Tap for more steps...

Step 2.3.4.1

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

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

Step 2.3.4.2

Multiply $5$ by $5$ .

$f'' (x) = \frac{25 \sin (5 x)}{3} \cdot \frac{d}{d x} (x)$

Step 2.3.5

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

$f'' (x) = \frac{25 \sin (5 x)}{3} \cdot 1$

Step 2.3.6

Multiply $\frac{25 \sin (5 x)}{3}$ by $1$ .

$f'' (x) = \frac{25 \sin (5 x)}{3}$

Step 3

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

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

Step 4

Set the numerator equal to zero.

$5 \cos (5 x) = 0$

Step 5

Solve the equation for $x$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Divide each term in $5 \cos (5 x) = 0$ by $5$ .

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

Step 5.1.2

Simplify the left side.

Tap for more steps...

Step 5.1.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.1.2.1.1

Cancel the common factor.

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

Step 5.1.2.1.2

Divide $\cos (5 x)$ by $1$ .

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

Step 5.1.3

Simplify the right side.

Tap for more steps...

Step 5.1.3.1

Divide $0$ by $5$ .

$\cos (5 x) = 0$

Step 5.2

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

$5 x = \arccos (0)$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

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

$5 x = \frac{π}{2}$

Step 5.4

Divide each term in $5 x = \frac{π}{2}$ by $5$ and simplify.

Tap for more steps...

Step 5.4.1

Divide each term in $5 x = \frac{π}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5}$

Step 5.4.2

Simplify the left side.

Tap for more steps...

Step 5.4.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{π}{2}}{5}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{5}$

Step 5.4.3

Simplify the right side.

Tap for more steps...

Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{5}$

Step 5.4.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{5}$ .

Tap for more steps...

Step 5.4.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{5}$ .

$x = \frac{π}{2 \cdot 5}$

Step 5.4.3.2.2

Multiply $2$ by $5$ .

$x = \frac{π}{10}$

Step 5.5

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.

$5 x = 2 π - \frac{π}{2}$

Step 5.6

Solve for $x$ .

Tap for more steps...

Step 5.6.1

Simplify.

Tap for more steps...

Step 5.6.1.1

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

$5 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 5.6.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$5 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 5.6.1.3

Combine the numerators over the common denominator.

$5 x = \frac{2 π \cdot 2 - π}{2}$

Step 5.6.1.4

Multiply $2$ by $2$ .

$5 x = \frac{4 π - π}{2}$

Step 5.6.1.5

Subtract $π$ from $4 π$ .

$5 x = \frac{3 π}{2}$

Step 5.6.2

Divide each term in $5 x = \frac{3 π}{2}$ by $5$ and simplify.

Tap for more steps...

Step 5.6.2.1

Divide each term in $5 x = \frac{3 π}{2}$ by $5$ .

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

Step 5.6.2.2

Simplify the left side.

Tap for more steps...

Step 5.6.2.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.6.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{\frac{3 π}{2}}{5}$

Step 5.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{5}$

Step 5.6.2.3

Simplify the right side.

Tap for more steps...

Step 5.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{5}$

Step 5.6.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{5}$ .

Tap for more steps...

Step 5.6.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{5}$ .

$x = \frac{3 π}{2 \cdot 5}$

Step 5.6.2.3.2.2

Multiply $2$ by $5$ .

$x = \frac{3 π}{10}$

Step 5.7

The solution to the equation $5 x = \frac{π}{2}$ .

$x = \frac{π}{10}, \frac{3 π}{10}$

Step 6

Evaluate the second derivative at $x = \frac{π}{10}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{25 \sin (5 (\frac{π}{10}))}{3}$

Step 7

Evaluate the second derivative.

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 7.1.1.1

Factor $5$ out of $10$ .

$\frac{25 \sin (5 \frac{π}{5 (2)})}{3}$

Step 7.1.1.2

Cancel the common factor.

$\frac{25 \sin (5 \frac{π}{5 \cdot 2})}{3}$

Step 7.1.1.3

Rewrite the expression.

$\frac{25 \sin (\frac{π}{2})}{3}$

Step 7.1.2

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

$\frac{25 \cdot 1}{3}$

Step 7.2

Multiply $25$ by $1$ .

$\frac{25}{3}$

Step 8

$x = \frac{π}{10}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{π}{10}$ is a local minimum

Step 9

Find the y-value when $x = \frac{π}{10}$ .

Tap for more steps...

Step 9.1

Replace the variable $x$ with $\frac{π}{10}$ in the expression.

$f (\frac{π}{10}) = 2 - \frac{\sin (5 (\frac{π}{10}))}{3}$

Step 9.2

Simplify the result.

Tap for more steps...

Step 9.2.1

Simplify the numerator.

Tap for more steps...

Step 9.2.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 9.2.1.1.1

Factor $5$ out of $10$ .

$f (\frac{π}{10}) = 2 - \frac{\sin (5 (\frac{π}{5 (2)}))}{3}$

Step 9.2.1.1.2

Cancel the common factor.

$f (\frac{π}{10}) = 2 - \frac{\sin (5 (\frac{π}{5 \cdot 2}))}{3}$

Step 9.2.1.1.3

Rewrite the expression.

$f (\frac{π}{10}) = 2 - \frac{\sin (\frac{π}{2})}{3}$

Step 9.2.1.2

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

$f (\frac{π}{10}) = 2 - \frac{1}{3}$

Step 9.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{π}{10}) = 2 \cdot \frac{3}{3} - \frac{1}{3}$

Step 9.2.3

Combine $2$ and $\frac{3}{3}$ .

$f (\frac{π}{10}) = \frac{2 \cdot 3}{3} - \frac{1}{3}$

Step 9.2.4

Combine the numerators over the common denominator.

$f (\frac{π}{10}) = \frac{2 \cdot 3 - 1}{3}$

Step 9.2.5

Simplify the numerator.

Tap for more steps...

Step 9.2.5.1

Multiply $2$ by $3$ .

$f (\frac{π}{10}) = \frac{6 - 1}{3}$

Step 9.2.5.2

Subtract $1$ from $6$ .

$f (\frac{π}{10}) = \frac{5}{3}$

Step 9.2.6

The final answer is $\frac{5}{3}$ .

$y = \frac{5}{3}$

Step 10

Evaluate the second derivative at $x = \frac{3 π}{10}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{25 \sin (5 (\frac{3 π}{10}))}{3}$

Step 11

Evaluate the second derivative.

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 11.1.1.1

Factor $5$ out of $10$ .

$\frac{25 \sin (5 \frac{3 π}{5 (2)})}{3}$

Step 11.1.1.2

Cancel the common factor.

$\frac{25 \sin (5 \frac{3 π}{5 \cdot 2})}{3}$

Step 11.1.1.3

Rewrite the expression.

$\frac{25 \sin (\frac{3 π}{2})}{3}$

Step 11.1.2

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.

$\frac{25 (- \sin (\frac{π}{2}))}{3}$

Step 11.1.3

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

$\frac{25 (- 1 \cdot 1)}{3}$

Step 11.1.4

Multiply $- 1$ by $1$ .

$\frac{25 \cdot - 1}{3}$

Step 11.2

Simplify the expression.

Tap for more steps...

Step 11.2.1

Multiply $25$ by $- 1$ .

$\frac{- 25}{3}$

Step 11.2.2

Move the negative in front of the fraction.

$- \frac{25}{3}$

Step 12

$x = \frac{3 π}{10}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3 π}{10}$ is a local maximum

Step 13

Find the y-value when $x = \frac{3 π}{10}$ .

Tap for more steps...

Step 13.1

Replace the variable $x$ with $\frac{3 π}{10}$ in the expression.

$f (\frac{3 π}{10}) = 2 - \frac{\sin (5 (\frac{3 π}{10}))}{3}$

Step 13.2

Simplify the result.

Tap for more steps...

Step 13.2.1

Simplify each term.

Tap for more steps...

Step 13.2.1.1

Simplify the numerator.

Tap for more steps...

Step 13.2.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 13.2.1.1.1.1

Factor $5$ out of $10$ .

$f (\frac{3 π}{10}) = 2 - \frac{\sin (5 (\frac{3 π}{5 (2)}))}{3}$

Step 13.2.1.1.1.2

Cancel the common factor.

$f (\frac{3 π}{10}) = 2 - \frac{\sin (5 (\frac{3 π}{5 \cdot 2}))}{3}$

Step 13.2.1.1.1.3

Rewrite the expression.

$f (\frac{3 π}{10}) = 2 - \frac{\sin (\frac{3 π}{2})}{3}$

Step 13.2.1.1.2

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.

$f (\frac{3 π}{10}) = 2 - \frac{- \sin (\frac{π}{2})}{3}$

Step 13.2.1.1.3

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

$f (\frac{3 π}{10}) = 2 - \frac{- 1 \cdot 1}{3}$

Step 13.2.1.1.4

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{10}) = 2 - \frac{- 1}{3}$

Step 13.2.1.2

Move the negative in front of the fraction.

$f (\frac{3 π}{10}) = 2 + \frac{1}{3}$

Step 13.2.1.3

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

Tap for more steps...

Step 13.2.1.3.1

Multiply $- 1$ by $- 1$ .

$f (\frac{3 π}{10}) = 2 + 1 (\frac{1}{3})$

Step 13.2.1.3.2

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

$f (\frac{3 π}{10}) = 2 + \frac{1}{3}$

Step 13.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{3 π}{10}) = 2 \cdot \frac{3}{3} + \frac{1}{3}$

Step 13.2.3

Combine $2$ and $\frac{3}{3}$ .

$f (\frac{3 π}{10}) = \frac{2 \cdot 3}{3} + \frac{1}{3}$

Step 13.2.4

Combine the numerators over the common denominator.

$f (\frac{3 π}{10}) = \frac{2 \cdot 3 + 1}{3}$

Step 13.2.5

Simplify the numerator.

Tap for more steps...

Step 13.2.5.1

Multiply $2$ by $3$ .

$f (\frac{3 π}{10}) = \frac{6 + 1}{3}$

Step 13.2.5.2

Add $6$ and $1$ .

$f (\frac{3 π}{10}) = \frac{7}{3}$

Step 13.2.6

The final answer is $\frac{7}{3}$ .

$y = \frac{7}{3}$

Step 14

These are the local extrema for $f (x) = 2 - \frac{\sin (5 x)}{3}$ .

$(\frac{π}{10}, \frac{5}{3})$ is a local minima

$(\frac{3 π}{10}, \frac{7}{3})$ is a local maxima

Step 15