Calculus Examples

$y = 6 \sin (π x - 6) - 6$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $6 \sin (π x - 6) - 6$ with respect to $x$ is $\frac{d}{d x} [6 \sin (π x - 6)] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [6 \sin (π x - 6)] + \frac{d}{d x} [- 6]$

Step 1.2

Evaluate $\frac{d}{d x} [6 \sin (π x - 6)]$ .

Tap for more steps...

Step 1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sin (π x - 6)$ with respect to $x$ is $6 \frac{d}{d x} [\sin (π x - 6)]$ .

$6 \frac{d}{d x} [\sin (π x - 6)] + \frac{d}{d x} [- 6]$

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 $g (x) = π x - 6$ .

Tap for more steps...

Step 1.2.2.1

To apply the Chain Rule, set $u$ as $π x - 6$ .

$6 (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [π x - 6]) + \frac{d}{d x} [- 6]$

Step 1.2.2.2

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

$6 (\cos (u) \frac{d}{d x} [π x - 6]) + \frac{d}{d x} [- 6]$

Step 1.2.2.3

Replace all occurrences of $u$ with $π x - 6$ .

$6 (\cos (π x - 6) \frac{d}{d x} [π x - 6]) + \frac{d}{d x} [- 6]$

Step 1.2.3

By the Sum Rule, the derivative of $π x - 6$ with respect to $x$ is $\frac{d}{d x} [π x] + \frac{d}{d x} [- 6]$ .

$6 (\cos (π x - 6) (\frac{d}{d x} [π x] + \frac{d}{d x} [- 6])) + \frac{d}{d x} [- 6]$

Step 1.2.4

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

$6 (\cos (π x - 6) (π \frac{d}{d x} [x] + \frac{d}{d x} [- 6])) + \frac{d}{d x} [- 6]$

Step 1.2.5

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

$6 (\cos (π x - 6) (π \cdot 1 + \frac{d}{d x} [- 6])) + \frac{d}{d x} [- 6]$

Step 1.2.6

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$6 (\cos (π x - 6) (π \cdot 1 + 0)) + \frac{d}{d x} [- 6]$

Step 1.2.7

Multiply $π$ by $1$ .

$6 (\cos (π x - 6) (π + 0)) + \frac{d}{d x} [- 6]$

Step 1.2.8

Add $π$ and $0$ .

$6 \cos (π x - 6) π + \frac{d}{d x} [- 6]$

Step 1.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$6 \cos (π x - 6) π + 0$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Add $6 \cos (π x - 6) π$ and $0$ .

$6 \cos (π x - 6) π$

Step 1.4.2

Reorder the factors of $6 \cos (π x - 6) π$ .

$6 π \cos (π x - 6)$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

Since $6 π$ is constant with respect to $x$ , the derivative of $6 π \cos (π x - 6)$ with respect to $x$ is $6 π \frac{d}{d x} [\cos (π x - 6)]$ .

$f'' (x) = 6 π \frac{d}{d x} (\cos (π x - 6))$

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 $g (x) = π x - 6$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $π x - 6$ .

$f'' (x) = 6 π (\frac{d}{d u} (\cos (u)) \frac{d}{d x} (π x - 6))$

Step 2.2.2

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

$f'' (x) = 6 π (- \sin (u) \frac{d}{d x} (π x - 6))$

Step 2.2.3

Replace all occurrences of $u$ with $π x - 6$ .

$f'' (x) = 6 π (- \sin (π x - 6) \frac{d}{d x} (π x - 6))$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Multiply $- 1$ by $6$ .

$f'' (x) = - 6 π (\sin (π x - 6) \frac{d}{d x} (π x - 6))$

Step 2.3.2

By the Sum Rule, the derivative of $π x - 6$ with respect to $x$ is $\frac{d}{d x} [π x] + \frac{d}{d x} [- 6]$ .

$f'' (x) = - 6 π \sin (π x - 6) (\frac{d}{d x} (π x) + \frac{d}{d x} (- 6))$

Step 2.3.3

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

$f'' (x) = - 6 π \sin (π x - 6) (π \frac{d}{d x} (x) + \frac{d}{d x} (- 6))$

Step 2.3.4

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

$f'' (x) = - 6 π \sin (π x - 6) (π \cdot 1 + \frac{d}{d x} (- 6))$

Step 2.3.5

Multiply $π$ by $1$ .

$f'' (x) = - 6 π \sin (π x - 6) (π + \frac{d}{d x} (- 6))$

Step 2.3.6

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$f'' (x) = - 6 π \sin (π x - 6) (π + 0)$

Step 2.3.7

Add $π$ and $0$ .

$f'' (x) = - 6 π \sin (π x - 6) π$

Step 2.4

Raise $π$ to the power of $1$ .

$f'' (x) = - 6 (π π) \sin (π x - 6)$

Step 2.5

Raise $π$ to the power of $1$ .

$f'' (x) = - 6 (π π) \sin (π x - 6)$

Step 2.6

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

$f'' (x) = - 6 π^{1 + 1} \sin (π x - 6)$

Step 2.7

Add $1$ and $1$ .

$f'' (x) = - 6 π^{2} \sin (π x - 6)$

Step 3

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

$6 π \cos (π x - 6) = 0$

Step 4

Divide each term in $6 π \cos (π x - 6) = 0$ by $6 π$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $6 π \cos (π x - 6) = 0$ by $6 π$ .

$\frac{6 π \cos (π x - 6)}{6 π} = \frac{0}{6 π}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{6 π \cos (π x - 6)}{6 π} = \frac{0}{6 π}$

Step 4.2.1.2

Rewrite the expression.

$\frac{π \cos (π x - 6)}{π} = \frac{0}{6 π}$

Step 4.2.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 4.2.2.1

Cancel the common factor.

$\frac{π \cos (π x - 6)}{π} = \frac{0}{6 π}$

Step 4.2.2.2

Divide $\cos (π x - 6)$ by $1$ .

$\cos (π x - 6) = \frac{0}{6 π}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Cancel the common factor of $0$ and $6$ .

Tap for more steps...

Step 4.3.1.1

Factor $6$ out of $0$ .

$\cos (π x - 6) = \frac{6 (0)}{6 π}$

Step 4.3.1.2

Cancel the common factors.

Tap for more steps...

Step 4.3.1.2.1

Factor $6$ out of $6 π$ .

$\cos (π x - 6) = \frac{6 (0)}{6 (π)}$

Step 4.3.1.2.2

Cancel the common factor.

$\cos (π x - 6) = \frac{6 \cdot 0}{6 π}$

Step 4.3.1.2.3

Rewrite the expression.

$\cos (π x - 6) = \frac{0}{π}$

Step 4.3.2

Divide $0$ by $π$ .

$\cos (π x - 6) = 0$

Step 5

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

$π x - 6 = \arccos (0)$

Step 6

Simplify the right side.

Tap for more steps...

Step 6.1

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

$π x - 6 = \frac{π}{2}$

Step 7

Add $6$ to both sides of the equation.

$π x = \frac{π}{2} + 6$

Step 8

Divide each term in $π x = \frac{π}{2} + 6$ by $π$ and simplify.

Tap for more steps...

Step 8.1

Divide each term in $π x = \frac{π}{2} + 6$ by $π$ .

$\frac{π x}{π} = \frac{\frac{π}{2}}{π} + \frac{6}{π}$

Step 8.2

Simplify the left side.

Tap for more steps...

Step 8.2.1

Cancel the common factor of $π$ .

Tap for more steps...

Step 8.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{\frac{π}{2}}{π} + \frac{6}{π}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{π} + \frac{6}{π}$

Step 8.3

Simplify the right side.

Tap for more steps...

Step 8.3.1

Simplify each term.

Tap for more steps...

Step 8.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{π} + \frac{6}{π}$

Step 8.3.1.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 8.3.1.2.1

Cancel the common factor.

$x = \frac{π}{2} \cdot \frac{1}{π} + \frac{6}{π}$

Step 8.3.1.2.2

Rewrite the expression.

$x = \frac{1}{2} + \frac{6}{π}$

Step 9

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.

$π x - 6 = 2 π - \frac{π}{2}$

Step 10

Solve for $x$ .

Tap for more steps...

Step 10.1

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 10.1.1

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

$π x - 6 = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 10.1.2

Combine fractions.

Tap for more steps...

Step 10.1.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$π x - 6 = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 10.1.2.2

Combine the numerators over the common denominator.

$π x - 6 = \frac{2 π \cdot 2 - π}{2}$

Step 10.1.3

Simplify the numerator.

Tap for more steps...

Step 10.1.3.1

Multiply $2$ by $2$ .

$π x - 6 = \frac{4 π - π}{2}$

Step 10.1.3.2

Subtract $π$ from $4 π$ .

$π x - 6 = \frac{3 π}{2}$

Step 10.2

Add $6$ to both sides of the equation.

$π x = \frac{3 π}{2} + 6$

Step 10.3

Divide each term in $π x = \frac{3 π}{2} + 6$ by $π$ and simplify.

Tap for more steps...

Step 10.3.1

Divide each term in $π x = \frac{3 π}{2} + 6$ by $π$ .

$\frac{π x}{π} = \frac{\frac{3 π}{2}}{π} + \frac{6}{π}$

Step 10.3.2

Simplify the left side.

Tap for more steps...

Step 10.3.2.1

Cancel the common factor of $π$ .

Tap for more steps...

Step 10.3.2.1.1

Cancel the common factor.

$\frac{π x}{π} = \frac{\frac{3 π}{2}}{π} + \frac{6}{π}$

Step 10.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{π} + \frac{6}{π}$

Step 10.3.3

Simplify the right side.

Tap for more steps...

Step 10.3.3.1

Simplify each term.

Tap for more steps...

Step 10.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{π} + \frac{6}{π}$

Step 10.3.3.1.2

Cancel the common factor of $π$ .

Tap for more steps...

Step 10.3.3.1.2.1

Factor $π$ out of $3 π$ .

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π} + \frac{6}{π}$

Step 10.3.3.1.2.2

Cancel the common factor.

$x = \frac{π \cdot 3}{2} \cdot \frac{1}{π} + \frac{6}{π}$

Step 10.3.3.1.2.3

Rewrite the expression.

$x = \frac{3}{2} + \frac{6}{π}$

Step 11

The solution to the equation $π x - 6 = \frac{π}{2}$ .

$x = \frac{1}{2} + \frac{6}{π}, \frac{3}{2} + \frac{6}{π}$

Step 12

Evaluate the second derivative at $x = \frac{1}{2} + \frac{6}{π}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 6 π^{2} \sin (π (\frac{1}{2} + \frac{6}{π}) - 6)$

Step 13

Evaluate the second derivative.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Apply the distributive property.

$- 6 π^{2} \sin (π \frac{1}{2} + π \frac{6}{π} - 6)$

Step 13.1.2

Combine $π$ and $\frac{1}{2}$ .

$- 6 π^{2} \sin (\frac{π}{2} + π \frac{6}{π} - 6)$

Step 13.1.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 13.1.3.1

Cancel the common factor.

$- 6 π^{2} \sin (\frac{π}{2} + π \frac{6}{π} - 6)$

Step 13.1.3.2

Rewrite the expression.

$- 6 π^{2} \sin (\frac{π}{2} + 6 - 6)$

Step 13.2

Simplify by subtracting numbers.

Tap for more steps...

Step 13.2.1

Subtract $6$ from $6$ .

$- 6 π^{2} \sin (\frac{π}{2} + 0)$

Step 13.2.2

Add $\frac{π}{2}$ and $0$ .

$- 6 π^{2} \sin (\frac{π}{2})$

Step 13.3

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

$- 6 π^{2} \cdot 1$

Step 13.4

Multiply $- 6$ by $1$ .

$- 6 π^{2}$

Step 14

$x = \frac{1}{2} + \frac{6}{π}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{1}{2} + \frac{6}{π}$ is a local maximum

Step 15

Find the y-value when $x = \frac{1}{2} + \frac{6}{π}$ .

Tap for more steps...

Step 15.1

Replace the variable $x$ with $\frac{1}{2} + \frac{6}{π}$ in the expression.

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (π (\frac{1}{2} + \frac{6}{π}) - 6) - 6$

Step 15.2

Simplify the result.

Tap for more steps...

Step 15.2.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1

Simplify each term.

Tap for more steps...

Step 15.2.1.1.1

Apply the distributive property.

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (π (\frac{1}{2}) + π (\frac{6}{π}) - 6) - 6$

Step 15.2.1.1.2

Combine $π$ and $\frac{1}{2}$ .

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (\frac{π}{2} + π (\frac{6}{π}) - 6) - 6$

Step 15.2.1.1.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 15.2.1.1.3.1

Cancel the common factor.

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (\frac{π}{2} + π (\frac{6}{π}) - 6) - 6$

Step 15.2.1.1.3.2

Rewrite the expression.

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (\frac{π}{2} + 6 - 6) - 6$

Step 15.2.1.2

Subtract $6$ from $6$ .

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (\frac{π}{2} + 0) - 6$

Step 15.2.1.3

Add $\frac{π}{2}$ and $0$ .

$f (\frac{1}{2} + \frac{6}{π}) = 6 \sin (\frac{π}{2}) - 6$

Step 15.2.1.4

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

$f (\frac{1}{2} + \frac{6}{π}) = 6 \cdot 1 - 6$

Step 15.2.1.5

Multiply $6$ by $1$ .

$f (\frac{1}{2} + \frac{6}{π}) = 6 - 6$

Step 15.2.2

Subtract $6$ from $6$ .

$f (\frac{1}{2} + \frac{6}{π}) = 0$

Step 15.2.3

The final answer is $0$ .

$y = 0$

Step 16

Evaluate the second derivative at $x = \frac{3}{2} + \frac{6}{π}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 6 π^{2} \sin (π (\frac{3}{2} + \frac{6}{π}) - 6)$

Step 17

Evaluate the second derivative.

Tap for more steps...

Step 17.1

Simplify each term.

Tap for more steps...

Step 17.1.1

Apply the distributive property.

$- 6 π^{2} \sin (π \frac{3}{2} + π \frac{6}{π} - 6)$

Step 17.1.2

Combine $π$ and $\frac{3}{2}$ .

$- 6 π^{2} \sin (\frac{π \cdot 3}{2} + π \frac{6}{π} - 6)$

Step 17.1.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 17.1.3.1

Cancel the common factor.

$- 6 π^{2} \sin (\frac{π \cdot 3}{2} + π \frac{6}{π} - 6)$

Step 17.1.3.2

Rewrite the expression.

$- 6 π^{2} \sin (\frac{π \cdot 3}{2} + 6 - 6)$

Step 17.1.4

Move $3$ to the left of $π$ .

$- 6 π^{2} \sin (\frac{3 π}{2} + 6 - 6)$

Step 17.2

Simplify by subtracting numbers.

Tap for more steps...

Step 17.2.1

Subtract $6$ from $6$ .

$- 6 π^{2} \sin (\frac{3 π}{2} + 0)$

Step 17.2.2

Add $\frac{3 π}{2}$ and $0$ .

$- 6 π^{2} \sin (\frac{3 π}{2})$

Step 17.3

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.

$- 6 π^{2} (- \sin (\frac{π}{2}))$

Step 17.4

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

$- 6 π^{2} (- 1 \cdot 1)$

Step 17.5

Multiply $- 1$ by $1$ .

$- 6 π^{2} \cdot - 1$

Step 17.6

Multiply $- 1$ by $- 6$ .

$6 π^{2}$

Step 18

$x = \frac{3}{2} + \frac{6}{π}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3}{2} + \frac{6}{π}$ is a local minimum

Step 19

Find the y-value when $x = \frac{3}{2} + \frac{6}{π}$ .

Tap for more steps...

Step 19.1

Replace the variable $x$ with $\frac{3}{2} + \frac{6}{π}$ in the expression.

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (π (\frac{3}{2} + \frac{6}{π}) - 6) - 6$

Step 19.2

Simplify the result.

Tap for more steps...

Step 19.2.1

Simplify each term.

Tap for more steps...

Step 19.2.1.1

Simplify each term.

Tap for more steps...

Step 19.2.1.1.1

Apply the distributive property.

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (π (\frac{3}{2}) + π (\frac{6}{π}) - 6) - 6$

Step 19.2.1.1.2

Combine $π$ and $\frac{3}{2}$ .

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{π \cdot 3}{2} + π (\frac{6}{π}) - 6) - 6$

Step 19.2.1.1.3

Cancel the common factor of $π$ .

Tap for more steps...

Step 19.2.1.1.3.1

Cancel the common factor.

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{π \cdot 3}{2} + π (\frac{6}{π}) - 6) - 6$

Step 19.2.1.1.3.2

Rewrite the expression.

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{π \cdot 3}{2} + 6 - 6) - 6$

Step 19.2.1.1.4

Move $3$ to the left of $π$ .

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{3 π}{2} + 6 - 6) - 6$

Step 19.2.1.2

Subtract $6$ from $6$ .

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{3 π}{2} + 0) - 6$

Step 19.2.1.3

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{3}{2} + \frac{6}{π}) = 6 \sin (\frac{3 π}{2}) - 6$

Step 19.2.1.4

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.

$f (\frac{3}{2} + \frac{6}{π}) = 6 (- \sin (\frac{π}{2})) - 6$

Step 19.2.1.5

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

$f (\frac{3}{2} + \frac{6}{π}) = 6 (- 1 \cdot 1) - 6$

Step 19.2.1.6

Multiply $6 (- 1 \cdot 1)$ .

Tap for more steps...

Step 19.2.1.6.1

Multiply $- 1$ by $1$ .

$f (\frac{3}{2} + \frac{6}{π}) = 6 \cdot - 1 - 6$

Step 19.2.1.6.2

Multiply $6$ by $- 1$ .

$f (\frac{3}{2} + \frac{6}{π}) = - 6 - 6$

Step 19.2.2

Subtract $6$ from $- 6$ .

$f (\frac{3}{2} + \frac{6}{π}) = - 12$

Step 19.2.3

The final answer is $- 12$ .

$y = - 12$

Step 20

These are the local extrema for $f (x) = 6 \sin (π x - 6) - 6$ .

$(\frac{1}{2} + \frac{6}{π}, 0)$ is a local maxima

$(\frac{3}{2} + \frac{6}{π}, - 12)$ is a local minima

Step 21