Calculus Examples

$y = - 13 \cos (h (x)) + 12 \sin (h (x))$

Step 1

Write $y = - 13 \cos (h (x)) + 12 \sin (h (x))$ as a function.

$f (x) = - 13 \cos (h (x)) + 12 \sin (h (x))$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- 13 \cos (h (x)) + 12 \sin (h (x))$ with respect to $x$ is $\frac{d}{d x} [- 13 \cos (h (x))] + \frac{d}{d x} [12 \sin (h (x))]$ .

$\frac{d}{d x} [- 13 \cos (h (x))] + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2

Evaluate $\frac{d}{d x} [- 13 \cos (h (x))]$ .

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Step 2.2.1

Since $- 13$ is constant with respect to $x$ , the derivative of $- 13 \cos (h x)$ with respect to $x$ is $- 13 \frac{d}{d x} [\cos (h x)]$ .

$- 13 \frac{d}{d x} [\cos (h x)] + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.2

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 $g (x) = h x$ .

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Step 2.2.2.1

To apply the Chain Rule, set $u_{1}$ as $h x$ .

$- 13 (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [h x]) + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- 13 (- \sin (h x) \frac{d}{d x} [h x]) + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.3

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

$- 13 (- \sin (h x) (h \frac{d}{d x} [x])) + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.4

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

$- 13 (- \sin (h x) (h \cdot 1)) + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.5

Multiply $h$ by $1$ .

$- 13 (- \sin (h x) h) + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.2.6

Multiply $- 1$ by $- 13$ .

$13 \sin (h x) h + \frac{d}{d x} [12 \sin (h (x))]$

Step 2.3

Evaluate $\frac{d}{d x} [12 \sin (h (x))]$ .

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Step 2.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 \sin (h x)$ with respect to $x$ is $12 \frac{d}{d x} [\sin (h x)]$ .

$13 \sin (h x) h + 12 \frac{d}{d x} [\sin (h x)]$

Step 2.3.2

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 $g (x) = h x$ .

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Step 2.3.2.1

To apply the Chain Rule, set $u_{2}$ as $h x$ .

$13 \sin (h x) h + 12 (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [h x])$

Step 2.3.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$13 \sin (h x) h + 12 (\cos (h x) \frac{d}{d x} [h x])$

Step 2.3.3

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

$13 \sin (h x) h + 12 (\cos (h x) (h \frac{d}{d x} [x]))$

Step 2.3.4

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

$13 \sin (h x) h + 12 (\cos (h x) (h \cdot 1))$

Step 2.3.5

Multiply $h$ by $1$ .

$13 \sin (h x) h + 12 \cos (h x) h$

Step 2.4

Reorder terms.

$13 h \sin (h x) + 12 h \cos (h x)$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $13 h \sin (h x) + 12 h \cos (h x)$ with respect to $x$ is $\frac{d}{d x} [13 h \sin (h x)] + \frac{d}{d x} [12 h \cos (h x)]$ .

$f'' (x) = \frac{d}{d x} (13 h \sin (h x)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2

Evaluate $\frac{d}{d x} [13 h \sin (h x)]$ .

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Step 3.2.1

Since $13 h$ is constant with respect to $x$ , the derivative of $13 h \sin (h x)$ with respect to $x$ is $13 h \frac{d}{d x} [\sin (h x)]$ .

$f'' (x) = 13 h \frac{d}{d x} (\sin (h x)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.2

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 $g (x) = h x$ .

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Step 3.2.2.1

To apply the Chain Rule, set $u_{1}$ as $h x$ .

$f'' (x) = 13 h (\frac{d}{d u (1)} (\sin (u_{1})) \frac{d}{d x} (h x)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.2.2

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

$f'' (x) = 13 h (\cos (u_{1}) \frac{d}{d x} (h x)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.2.3

Replace all occurrences of $u_{1}$ with $h x$ .

$f'' (x) = 13 h (\cos (h x) \frac{d}{d x} (h x)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.3

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

$f'' (x) = 13 h (\cos (h x) (h \frac{d}{d x} (x))) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.4

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

$f'' (x) = 13 h (\cos (h x) (h \cdot 1)) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.5

Multiply $h$ by $1$ .

$f'' (x) = 13 h (\cos (h x) h) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.6

Raise $h$ to the power of $1$ .

$f'' (x) = 13 (h \cdot h) \cos (h x) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.7

Raise $h$ to the power of $1$ .

$f'' (x) = 13 (h \cdot h) \cos (h x) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.8

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

$f'' (x) = 13 h^{1 + 1} \cos (h x) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.2.9

Add $1$ and $1$ .

$f'' (x) = 13 h^{2} \cos (h x) + \frac{d}{d x} (12 h \cos (h x))$

Step 3.3

Evaluate $\frac{d}{d x} [12 h \cos (h x)]$ .

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Step 3.3.1

Since $12 h$ is constant with respect to $x$ , the derivative of $12 h \cos (h x)$ with respect to $x$ is $12 h \frac{d}{d x} [\cos (h x)]$ .

$f'' (x) = 13 h^{2} \cos (h x) + 12 h \frac{d}{d x} (\cos (h x))$

Step 3.3.2

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 $g (x) = h x$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $h x$ .

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (\frac{d}{d u (2)} (\cos (u_{2})) \frac{d}{d x} (h x))$

Step 3.3.2.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (- \sin (u_{2}) \frac{d}{d x} (h x))$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $h x$ .

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (- \sin (h x) \frac{d}{d x} (h x))$

Step 3.3.3

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

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (- \sin (h x) (h \frac{d}{d x} (x)))$

Step 3.3.4

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

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (- \sin (h x) (h \cdot 1))$

Step 3.3.5

Multiply $h$ by $1$ .

$f'' (x) = 13 h^{2} \cos (h x) + 12 h (- \sin (h x) h)$

Step 3.3.6

Multiply $- 1$ by $12$ .

$f'' (x) = 13 h^{2} \cos (h x) - 12 h (\sin (h x) h)$

Step 3.3.7

Raise $h$ to the power of $1$ .

$f'' (x) = 13 h^{2} \cos (h x) - 12 (h \cdot h) \sin (h x)$

Step 3.3.8

Raise $h$ to the power of $1$ .

$f'' (x) = 13 h^{2} \cos (h x) - 12 (h \cdot h) \sin (h x)$

Step 3.3.9

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

$f'' (x) = 13 h^{2} \cos (h x) - 12 h^{1 + 1} \sin (h x)$

Step 3.3.10

Add $1$ and $1$ .

$f'' (x) = 13 h^{2} \cos (h x) - 12 h^{2} \sin (h x)$

Step 4

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

$13 h \sin (h x) + 12 h \cos (h x) = 0$

Step 5

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

$\frac{13 h \sin (h x)}{\cos (h x)} + \frac{12 h \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6

Separate fractions.

$\frac{13 h}{1} \cdot \frac{\sin (h x)}{\cos (h x)} + \frac{12 h \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 7

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

$\frac{13 h}{1} \cdot \tan (h x) + \frac{12 h \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 8

Divide $13 h$ by $1$ .

$13 h \tan (h x) + \frac{12 h \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 9

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

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Step 9.1

Cancel the common factor.

$13 h \tan (h x) + \frac{12 h \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 9.2

Divide $12 h$ by $1$ .

$13 h \tan (h x) + 12 h = \frac{0}{\cos (h x)}$

Step 10

Separate fractions.

$13 h \tan (h x) + 12 h = \frac{0}{1} \cdot \frac{1}{\cos (h x)}$

Step 11

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

$13 h \tan (h x) + 12 h = \frac{0}{1} \cdot \sec (h x)$

Step 12

Divide $0$ by $1$ .

$13 h \tan (h x) + 12 h = 0 \sec (h x)$

Step 13

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

$13 h \tan (h x) + 12 h = 0$

Step 14

Subtract $12 h$ from both sides of the equation.

$13 h \tan (h x) = - 12 h$

Step 15

Divide each term in $13 h \tan (h x) = - 12 h$ by $13 h$ and simplify.

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Step 15.1

Divide each term in $13 h \tan (h x) = - 12 h$ by $13 h$ .

$\frac{13 h \tan (h x)}{13 h} = \frac{- 12 h}{13 h}$

Step 15.2

Simplify the left side.

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Step 15.2.1

Cancel the common factor of $13$ .

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Step 15.2.1.1

Cancel the common factor.

$\frac{13 h \tan (h x)}{13 h} = \frac{- 12 h}{13 h}$

Step 15.2.1.2

Rewrite the expression.

$\frac{h \tan (h x)}{h} = \frac{- 12 h}{13 h}$

Step 15.2.2

Cancel the common factor of $h$ .

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Step 15.2.2.1

Cancel the common factor.

$\frac{h \tan (h x)}{h} = \frac{- 12 h}{13 h}$

Step 15.2.2.2

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

$\tan (h x) = \frac{- 12 h}{13 h}$

Step 15.3

Simplify the right side.

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Step 15.3.1

Cancel the common factor of $h$ .

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Step 15.3.1.1

Cancel the common factor.

$\tan (h x) = \frac{- 12 h}{13 h}$

Step 15.3.1.2

Rewrite the expression.

$\tan (h x) = \frac{- 12}{13}$

Step 15.3.2

Move the negative in front of the fraction.

$\tan (h x) = - \frac{12}{13}$

Step 16

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

$h x = \arctan (- \frac{12}{13})$

Step 17

Simplify the right side.

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Step 17.1

Evaluate $\arctan (- \frac{12}{13})$ .

$h x = - 0.74541947$

Step 18

Divide each term in $h x = - 0.74541947$ by $h$ and simplify.

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Step 18.1

Divide each term in $h x = - 0.74541947$ by $h$ .

$\frac{h x}{h} = \frac{- 0.74541947}{h}$

Step 18.2

Simplify the left side.

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Step 18.2.1

Cancel the common factor of $h$ .

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Step 18.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{- 0.74541947}{h}$

Step 18.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.74541947}{h}$

Step 18.3

Simplify the right side.

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Step 18.3.1

Move the negative in front of the fraction.

$x = - \frac{0.74541947}{h}$

Step 19

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

$h x = - 0.74541947 - (3.14159265)$

Step 20

Add $2 π$ to $- 0.74541947 - (3.14159265)$ .

$h x = - 0.74541947 - (3.14159265) + 2 π$

Step 21

The resulting angle of $2.39617317$ is positive and coterminal with $- 0.74541947 - (3.14159265)$ .

$h x = 2.39617317$

Step 22

Divide each term in $h x = 2.39617317$ by $h$ and simplify.

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Step 22.1

Divide each term in $h x = 2.39617317$ by $h$ .

$\frac{h x}{h} = \frac{2.39617317}{h}$

Step 22.2

Simplify the left side.

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Step 22.2.1

Cancel the common factor of $h$ .

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Step 22.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{2.39617317}{h}$

Step 22.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.39617317}{h}$

Step 23

Evaluate the second derivative at $x = \frac{2.39617317}{h}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$13 h^{2} \cos (h (\frac{2.39617317}{h})) - 12 h^{2} \sin (h (\frac{2.39617317}{h}))$

Step 24

Simplify each term.

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Step 24.1

Cancel the common factor of $h$ .

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Step 24.1.1

Cancel the common factor.

$13 h^{2} \cos (h \frac{2.39617317}{h}) - 12 h^{2} \sin (h (\frac{2.39617317}{h}))$

Step 24.1.2

Rewrite the expression.

$13 h^{2} \cos (2.39617317) - 12 h^{2} \sin (h (\frac{2.39617317}{h}))$

Step 24.2

Cancel the common factor of $h$ .

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Step 24.2.1

Cancel the common factor.

$13 h^{2} \cos (2.39617317) - 12 h^{2} \sin (h \frac{2.39617317}{h})$

Step 24.2.2

Rewrite the expression.

$13 h^{2} \cos (2.39617317) - 12 h^{2} \sin (2.39617317)$

Step 25

Since the first derivative test failed, there are no local extrema.

No Local Extrema

Step 26