Calculus Examples

$f (x) = \cos (x) + 2 \sin (x)$

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

$- \sin (x) + 2 \cos (x)$

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

$f'' (x) = - \cos (x) + 2 (- \sin (x))$

Step 2.3.3

Multiply $- 1$ by $2$ .

$f'' (x) = - \cos (x) - 2 \sin (x)$

Step 3

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

$- \sin (x) + 2 \cos (x) = 0$

Step 4

Divide each term in the equation by $\cos (x)$ .

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

Step 5

Separate fractions.

$\frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{- 1}{1} \cdot \tan (x) + \frac{2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7

Divide $- 1$ by $1$ .

$- \tan (x) + \frac{2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 8

Cancel the common factor of $\cos (x)$ .

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Step 8.1

Cancel the common factor.

$- \tan (x) + \frac{2 \cos (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 8.2

Divide $2$ by $1$ .

$- \tan (x) + 2 = \frac{0}{\cos (x)}$

Step 9

Separate fractions.

$- \tan (x) + 2 = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \tan (x) + 2 = \frac{0}{1} \cdot \sec (x)$

Step 11

Divide $0$ by $1$ .

$- \tan (x) + 2 = 0 \sec (x)$

Step 12

Multiply $0$ by $\sec (x)$ .

$- \tan (x) + 2 = 0$

Step 13

Subtract $2$ from both sides of the equation.

$- \tan (x) = - 2$

Step 14

Divide each term in $- \tan (x) = - 2$ by $- 1$ and simplify.

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Step 14.1

Divide each term in $- \tan (x) = - 2$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 2}{- 1}$

Step 14.2

Simplify the left side.

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Step 14.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 2}{- 1}$

Step 14.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- 2}{- 1}$

Step 14.3

Simplify the right side.

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Step 14.3.1

Divide $- 2$ by $- 1$ .

$\tan (x) = 2$

Step 15

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

$x = \arctan (2)$

Step 16

Simplify the right side.

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Step 16.1

Evaluate $\arctan (2)$ .

$x = 1.10714871$

Step 17

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 1.10714871$

Step 18

Solve for $x$ .

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Step 18.1

Remove parentheses.

$x = 3.14159265 + 1.10714871$

Step 18.2

Remove parentheses.

$x = (3.14159265) + 1.10714871$

Step 18.3

Add $3.14159265$ and $1.10714871$ .

$x = 4.24874137$

Step 19

The solution to the equation $x = 1.10714871$ .

$x = 1.10714871, 4.24874137$

Step 20

Evaluate the second derivative at $x = 1.10714871$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \cos (1.10714871) - 2 \sin (1.10714871)$

Step 21

$x = 1.10714871$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 1.10714871$ is a local maximum

Step 22

Find the y-value when $x = 1.10714871$ .

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Step 22.1

Replace the variable $x$ with $1.10714871$ in the expression.

$f (1.10714871) = \cos (1.10714871) + 2 \sin (1.10714871)$

Step 22.2

The final answer is $\cos (1.10714871) + 2 \sin (1.10714871)$ .

$y = \cos (1.10714871) + 2 \sin (1.10714871)$

Step 23

Evaluate the second derivative at $x = 4.24874137$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \cos (4.24874137) - 2 \sin (4.24874137)$

Step 24

$x = 4.24874137$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 4.24874137$ is a local minimum

Step 25

Find the y-value when $x = 4.24874137$ .

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Step 25.1

Replace the variable $x$ with $4.24874137$ in the expression.

$f (4.24874137) = \cos (4.24874137) + 2 \sin (4.24874137)$

Step 25.2

The final answer is $\cos (4.24874137) + 2 \sin (4.24874137)$ .

$y = \cos (4.24874137) + 2 \sin (4.24874137)$

Step 26

These are the local extrema for $f (x) = \cos (x) + 2 \sin (x)$ .

$(1.10714871, \cos (1.10714871) + 2 \sin (1.10714871))$ is a local maxima

$(4.24874137, \cos (4.24874137) + 2 \sin (4.24874137))$ is a local minima

Step 27