Calculus Examples

$f (x) = 4 \cos (\frac{x}{2}) - 3$

Step 1

Find the first derivative of the function.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $4 \cos (\frac{x}{2}) - 3$ with respect to $x$ is $\frac{d}{d x} [4 \cos (\frac{x}{2})] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [4 \cos (\frac{x}{2})] + \frac{d}{d x} [- 3]$

Step 1.2

Evaluate $\frac{d}{d x} [4 \cos (\frac{x}{2})]$ .

Tap for more steps...

Step 1.2.1

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

$4 \frac{d}{d x} [\cos (\frac{x}{2})] + \frac{d}{d x} [- 3]$

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 $f (x) = \cos (x)$ and $g (x) = \frac{x}{2}$ .

Tap for more steps...

Step 1.2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [- 3]$

Step 1.2.2.2

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

$4 (- \sin (u) \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [- 3]$

Step 1.2.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$4 (- \sin (\frac{x}{2}) \frac{d}{d x} [\frac{x}{2}]) + \frac{d}{d x} [- 3]$

Step 1.2.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$4 (- \sin (\frac{x}{2}) (\frac{1}{2} \frac{d}{d x} [x])) + \frac{d}{d x} [- 3]$

Step 1.2.4

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

$4 (- \sin (\frac{x}{2}) (\frac{1}{2} \cdot 1)) + \frac{d}{d x} [- 3]$

Step 1.2.5

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

$4 (- \sin (\frac{x}{2}) \frac{1}{2}) + \frac{d}{d x} [- 3]$

Step 1.2.6

Combine $\frac{1}{2}$ and $\sin (\frac{x}{2})$ .

$4 (- \frac{\sin (\frac{x}{2})}{2}) + \frac{d}{d x} [- 3]$

Step 1.2.7

Multiply $- 1$ by $4$ .

$- 4 \frac{\sin (\frac{x}{2})}{2} + \frac{d}{d x} [- 3]$

Step 1.2.8

Combine $- 4$ and $\frac{\sin (\frac{x}{2})}{2}$ .

$\frac{- 4 \sin (\frac{x}{2})}{2} + \frac{d}{d x} [- 3]$

Step 1.2.9

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 1.2.9.1

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

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

Step 1.2.9.2

Cancel the common factors.

Tap for more steps...

Step 1.2.9.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 2 \sin (\frac{x}{2}))}{2 (1)} + \frac{d}{d x} [- 3]$

Step 1.2.9.2.2

Cancel the common factor.

$\frac{2 (- 2 \sin (\frac{x}{2}))}{2 \cdot 1} + \frac{d}{d x} [- 3]$

Step 1.2.9.2.3

Rewrite the expression.

$\frac{- 2 \sin (\frac{x}{2})}{1} + \frac{d}{d x} [- 3]$

Step 1.2.9.2.4

Divide $- 2 \sin (\frac{x}{2})$ by $1$ .

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

Step 1.3

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$- 2 \sin (\frac{x}{2}) + 0$

Step 1.3.2

Add $- 2 \sin (\frac{x}{2})$ and $0$ .

$- 2 \sin (\frac{x}{2})$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

$f'' (x) = - 2 \frac{d}{d x} \sin (\frac{x}{2})$

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 $g (x) = \frac{x}{2}$ .

Tap for more steps...

Step 2.2.1

To apply the Chain Rule, set $u$ as $\frac{x}{2}$ .

$f'' (x) = - 2 (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (\frac{x}{2}))$

Step 2.2.2

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

$f'' (x) = - 2 (\cos (u) \frac{d}{d x} (\frac{x}{2}))$

Step 2.2.3

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$f'' (x) = - 2 (\cos (\frac{x}{2}) \frac{d}{d x} (\frac{x}{2}))$

Step 2.3

Differentiate.

Tap for more steps...

Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$f'' (x) = - 2 \cos (\frac{x}{2}) (\frac{1}{2} \cdot \frac{d}{d x} (x))$

Step 2.3.2

Simplify terms.

Tap for more steps...

Step 2.3.2.1

Combine $\frac{1}{2}$ and $- 2$ .

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

Step 2.3.2.2

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

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

Step 2.3.2.3

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 2.3.2.3.1

Factor $2$ out of $- 2 \cos (\frac{x}{2})$ .

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

Step 2.3.2.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.3.2.1

Factor $2$ out of $2$ .

$f'' (x) = \frac{2 (- \cos (\frac{x}{2}))}{2 (1)} \cdot \frac{d}{d x} (x)$

Step 2.3.2.3.2.2

Cancel the common factor.

$f'' (x) = \frac{2 (- \cos (\frac{x}{2}))}{2 \cdot 1} \cdot \frac{d}{d x} (x)$

Step 2.3.2.3.2.3

Rewrite the expression.

$f'' (x) = \frac{- \cos (\frac{x}{2})}{1} \cdot \frac{d}{d x} (x)$

Step 2.3.2.3.2.4

Divide $- \cos (\frac{x}{2})$ by $1$ .

$f'' (x) = - \cos (\frac{x}{2}) \frac{d}{d x} x$

Step 2.3.3

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

$f'' (x) = - \cos (\frac{x}{2}) \cdot 1$

Step 2.3.4

Multiply $- 1$ by $1$ .

$f'' (x) = - \cos (\frac{x}{2})$

Step 3

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

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

Step 4

Divide each term in $- 2 \sin (\frac{x}{2}) = 0$ by $- 2$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $- 2 \sin (\frac{x}{2}) = 0$ by $- 2$ .

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

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $\sin (\frac{x}{2})$ by $1$ .

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

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $0$ by $- 2$ .

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

Step 5

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

$\frac{x}{2} = \arcsin (0)$

Step 6

Simplify the right side.

Tap for more steps...

Step 6.1

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

$\frac{x}{2} = 0$

Step 7

Set the numerator equal to zero.

$x = 0$

Step 8

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.

$\frac{x}{2} = π - 0$

Step 9

Solve for $x$ .

Tap for more steps...

Step 9.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π - 0)$

Step 9.2

Simplify both sides of the equation.

Tap for more steps...

Step 9.2.1

Simplify the left side.

Tap for more steps...

Step 9.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π - 0)$

Step 9.2.1.1.2

Rewrite the expression.

$x = 2 (π - 0)$

Step 9.2.2

Simplify the right side.

Tap for more steps...

Step 9.2.2.1

Subtract $0$ from $π$ .

$x = 2 π$

Step 10

The solution to the equation $\frac{x}{2} = 0$ .

$x = 0, 2 π$

Step 11

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.

$- \cos (\frac{0}{2})$

Step 12

Evaluate the second derivative.

Tap for more steps...

Step 12.1

Divide $0$ by $2$ .

$- \cos (0)$

Step 12.2

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

$- 1 \cdot 1$

Step 12.3

Multiply $- 1$ by $1$ .

$- 1$

Step 13

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

$x = 0$ is a local maximum

Step 14

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

Tap for more steps...

Step 14.1

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

$f (0) = 4 \cos (\frac{0}{2}) - 3$

Step 14.2

Simplify the result.

Tap for more steps...

Step 14.2.1

Simplify each term.

Tap for more steps...

Step 14.2.1.1

Divide $0$ by $2$ .

$f (0) = 4 \cos (0) - 3$

Step 14.2.1.2

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

$f (0) = 4 \cdot 1 - 3$

Step 14.2.1.3

Multiply $4$ by $1$ .

$f (0) = 4 - 3$

Step 14.2.2

Subtract $3$ from $4$ .

$f (0) = 1$

Step 14.2.3

The final answer is $1$ .

$y = 1$

Step 15

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

$- \cos (\frac{2 π}{2})$

Step 16

Evaluate the second derivative.

Tap for more steps...

Step 16.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 16.1.1

Cancel the common factor.

$- \cos (\frac{2 π}{2})$

Step 16.1.2

Divide $π$ by $1$ .

$- \cos (π)$

Step 16.2

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.

$- - \cos (0)$

Step 16.3

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

$- (- 1 \cdot 1)$

Step 16.4

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

Tap for more steps...

Step 16.4.1

Multiply $- 1$ by $1$ .

$- - 1$

Step 16.4.2

Multiply $- 1$ by $- 1$ .

$1$

Step 17

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

$x = 2 π$ is a local minimum

Step 18

Find the y-value when $x = 2 π$ .

Tap for more steps...

Step 18.1

Replace the variable $x$ with $2 π$ in the expression.

$f (2 π) = 4 \cos (\frac{2 π}{2}) - 3$

Step 18.2

Simplify the result.

Tap for more steps...

Step 18.2.1

Simplify each term.

Tap for more steps...

Step 18.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.2.1.1.1

Cancel the common factor.

$f (2 π) = 4 \cos (\frac{2 π}{2}) - 3$

Step 18.2.1.1.2

Divide $π$ by $1$ .

$f (2 π) = 4 \cos (π) - 3$

Step 18.2.1.2

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.

$f (2 π) = 4 (- \cos (0)) - 3$

Step 18.2.1.3

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

$f (2 π) = 4 (- 1 \cdot 1) - 3$

Step 18.2.1.4

Multiply $4 (- 1 \cdot 1)$ .

Tap for more steps...

Step 18.2.1.4.1

Multiply $- 1$ by $1$ .

$f (2 π) = 4 \cdot - 1 - 3$

Step 18.2.1.4.2

Multiply $4$ by $- 1$ .

$f (2 π) = - 4 - 3$

Step 18.2.2

Subtract $3$ from $- 4$ .

$f (2 π) = - 7$

Step 18.2.3

The final answer is $- 7$ .

$y = - 7$

Step 19

These are the local extrema for $f (x) = 4 \cos (\frac{x}{2}) - 3$ .

$(0, 1)$ is a local maxima

$(2 π, - 7)$ is a local minima

Step 20