Calculus Examples

$y = \cos (x + \frac{π}{4})$

Step 1

Find the first derivative of the function.

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Step 1.1

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 $g (x) = x + \frac{π}{4}$ .

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Step 1.1.1

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

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x + \frac{π}{4}]$

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x + \frac{π}{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{4}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Since $\frac{π}{4}$ is constant with respect to $x$ , the derivative of $\frac{π}{4}$ with respect to $x$ is $0$ .

$- \sin (x + \frac{π}{4}) (1 + 0)$

Step 1.2.4

Simplify the expression.

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Step 1.2.4.1

Add $1$ and $0$ .

$- \sin (x + \frac{π}{4}) \cdot 1$

Step 1.2.4.2

Multiply $- 1$ by $1$ .

$- \sin (x + \frac{π}{4})$

Step 2

Find the second derivative of the function.

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Step 2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x + \frac{π}{4})$ with respect to $x$ is $- \frac{d}{d x} [\sin (x + \frac{π}{4})]$ .

$f'' (x) = - \frac{d}{d x} \sin (x + \frac{π}{4})$

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 $g (x) = x + \frac{π}{4}$ .

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Step 2.2.1

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

$f'' (x) = - (\frac{d}{d u} (\sin (u)) \frac{d}{d x} (x + \frac{π}{4}))$

Step 2.2.2

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

$f'' (x) = - (\cos (u) \frac{d}{d x} (x + \frac{π}{4}))$

Step 2.2.3

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

$f'' (x) = - (\cos (x + \frac{π}{4}) \frac{d}{d x} (x + \frac{π}{4}))$

Step 2.3

Differentiate.

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Step 2.3.1

By the Sum Rule, the derivative of $x + \frac{π}{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{π}{4}]$ .

$f'' (x) = - \cos (x + \frac{π}{4}) (\frac{d}{d x} (x) + \frac{d}{d x} (\frac{π}{4}))$

Step 2.3.2

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

$f'' (x) = - \cos (x + \frac{π}{4}) (1 + \frac{d}{d x} (\frac{π}{4}))$

Step 2.3.3

Since $\frac{π}{4}$ is constant with respect to $x$ , the derivative of $\frac{π}{4}$ with respect to $x$ is $0$ .

$f'' (x) = - \cos (x + \frac{π}{4}) (1 + 0)$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

Add $1$ and $0$ .

$f'' (x) = - \cos (x + \frac{π}{4}) \cdot 1$

Step 2.3.4.2

Multiply $- 1$ by $1$ .

$f'' (x) = - \cos (x + \frac{π}{4})$

Step 3

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

$- \sin (x + \frac{π}{4}) = 0$

Step 4

Divide each term in $- \sin (x + \frac{π}{4}) = 0$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \sin (x + \frac{π}{4}) = 0$ by $- 1$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

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

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 1$ .

$\sin (x + \frac{π}{4}) = 0$

Step 5

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

$x + \frac{π}{4} = \arcsin (0)$

Step 6

Simplify the right side.

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Step 6.1

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

$x + \frac{π}{4} = 0$

Step 7

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = - \frac{π}{4}$

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.

$x + \frac{π}{4} = π - 0$

Step 9

Solve for $x$ .

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Step 9.1

Subtract $0$ from $π$ .

$x + \frac{π}{4} = π$

Step 9.2

Move all terms not containing $x$ to the right side of the equation.

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Step 9.2.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$x = π - \frac{π}{4}$

Step 9.2.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 9.2.3

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 9.2.4

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 - π}{4}$

Step 9.2.5

Simplify the numerator.

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Step 9.2.5.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π - π}{4}$

Step 9.2.5.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{4}$

Step 10

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

$x = - \frac{π}{4}, \frac{3 π}{4}$

Step 11

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

$- \cos ((- \frac{π}{4}) + \frac{π}{4})$

Step 12

Evaluate the second derivative.

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Step 12.1

Combine the numerators over the common denominator.

$- \cos (\frac{- π + π}{4})$

Step 12.2

Add $- π$ and $π$ .

$- \cos (\frac{0}{4})$

Step 12.3

Divide $0$ by $4$ .

$- \cos (0)$

Step 12.4

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

$- 1 \cdot 1$

Step 12.5

Multiply $- 1$ by $1$ .

$- 1$

Step 13

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

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

Step 14

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

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Step 14.1

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

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

Step 14.2

Simplify the result.

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Step 14.2.1

Combine the numerators over the common denominator.

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

Step 14.2.2

Add $- π$ and $π$ .

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

Step 14.2.3

Divide $0$ by $4$ .

$f (- \frac{π}{4}) = \cos (0)$

Step 14.2.4

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

$f (- \frac{π}{4}) = 1$

Step 14.2.5

The final answer is $1$ .

$y = 1$

Step 15

Evaluate the second derivative at $x = \frac{3 π}{4}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \cos ((\frac{3 π}{4}) + \frac{π}{4})$

Step 16

Evaluate the second derivative.

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Step 16.1

Combine the numerators over the common denominator.

$- \cos (\frac{3 π + π}{4})$

Step 16.2

Add $3 π$ and $π$ .

$- \cos (\frac{4 π}{4})$

Step 16.3

Cancel the common factor of $4$ .

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Step 16.3.1

Cancel the common factor.

$- \cos (\frac{4 π}{4})$

Step 16.3.2

Divide $π$ by $1$ .

$- \cos (π)$

Step 16.4

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 16.5

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

$- (- 1 \cdot 1)$

Step 16.6

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

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Step 16.6.1

Multiply $- 1$ by $1$ .

$- - 1$

Step 16.6.2

Multiply $- 1$ by $- 1$ .

$1$

Step 17

$x = \frac{3 π}{4}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{3 π}{4}$ is a local minimum

Step 18

Find the y-value when $x = \frac{3 π}{4}$ .

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Step 18.1

Replace the variable $x$ with $\frac{3 π}{4}$ in the expression.

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

Step 18.2

Simplify the result.

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Step 18.2.1

Combine the numerators over the common denominator.

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

Step 18.2.2

Add $3 π$ and $π$ .

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

Step 18.2.3

Cancel the common factor of $4$ .

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Step 18.2.3.1

Cancel the common factor.

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

Step 18.2.3.2

Divide $π$ by $1$ .

$f (\frac{3 π}{4}) = \cos (π)$

Step 18.2.4

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.

$f (\frac{3 π}{4}) = - \cos (0)$

Step 18.2.5

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

$f (\frac{3 π}{4}) = - 1 \cdot 1$

Step 18.2.6

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{4}) = - 1$

Step 18.2.7

The final answer is $- 1$ .

$y = - 1$

Step 19

These are the local extrema for $f (x) = \cos (x + \frac{π}{4})$ .

$(- \frac{π}{4}, 1)$ is a local maxima

$(\frac{3 π}{4}, - 1)$ is a local minima

Step 20