Calculus Examples

$y = - \frac{145}{2} \cdot \cos (h (x)) + \frac{143}{2} \cdot \sin (h (x))$

Step 1

Write $y = - \frac{145}{2} \cdot \cos (h (x)) + \frac{143}{2} \cdot \sin (h (x))$ as a function.

$f (x) = - \frac{145}{2} \cdot \cos (h (x)) + \frac{143}{2} \cdot \sin (h (x))$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{145}{2} \cdot \cos (h (x)) + \frac{143}{2} \cdot \sin (h (x))$ with respect to $x$ is $\frac{d}{d x} [- \frac{145}{2} \cdot \cos (h (x))] + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$ .

$\frac{d}{d x} [- \frac{145}{2} \cdot \cos (h (x))] + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{145}{2} \cdot \cos (h (x))]$ .

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Step 2.2.1

Since $- \frac{145}{2}$ is constant with respect to $x$ , the derivative of $- \frac{145}{2} \cdot \cos (h x)$ with respect to $x$ is $- \frac{145}{2} \frac{d}{d x} [\cos (h x)]$ .

$- \frac{145}{2} \frac{d}{d x} [\cos (h x)] + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

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 $h x$ .

$- \frac{145}{2} (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [h x]) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.2.2

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

$- \frac{145}{2} (- \sin (h x) \frac{d}{d x} [h x]) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.3

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

$- \frac{145}{2} (- \sin (h x) (h \frac{d}{d x} [x])) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.4

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

$- \frac{145}{2} (- \sin (h x) (h \cdot 1)) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.5

Multiply $h$ by $1$ .

$- \frac{145}{2} (- \sin (h x) h) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.6

Multiply $- 1$ by $- 1$ .

$1 (\frac{145}{2}) (\sin (h x) h) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.7

Multiply $\frac{145}{2}$ by $1$ .

$\frac{145}{2} (\sin (h x) h) + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.8

Combine $\sin (h x)$ and $\frac{145}{2}$ .

$\frac{\sin (h x) \cdot 145}{2} h + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.9

Combine $\frac{\sin (h x) \cdot 145}{2}$ and $h$ .

$\frac{\sin (h x) \cdot 145 h}{2} + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.2.10

Move $145$ to the left of $\sin (h x)$ .

$\frac{145 \sin (h x) h}{2} + \frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{143}{2} \cdot \sin (h (x))]$ .

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Step 2.3.1

Since $\frac{143}{2}$ is constant with respect to $x$ , the derivative of $\frac{143}{2} \cdot \sin (h x)$ with respect to $x$ is $\frac{143}{2} \frac{d}{d x} [\sin (h x)]$ .

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} \frac{d}{d x} [\sin (h x)]$

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 $h x$ .

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [h x])$

Step 2.3.2.2

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

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} (\cos (h x) \frac{d}{d x} [h x])$

Step 2.3.3

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

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} (\cos (h x) (h \frac{d}{d x} [x]))$

Step 2.3.4

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

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} (\cos (h x) (h \cdot 1))$

Step 2.3.5

Multiply $h$ by $1$ .

$\frac{145 \sin (h x) h}{2} + \frac{143}{2} (\cos (h x) h)$

Step 2.3.6

Combine $\cos (h x)$ and $\frac{143}{2}$ .

$\frac{145 \sin (h x) h}{2} + \frac{\cos (h x) \cdot 143}{2} h$

Step 2.3.7

Combine $\frac{\cos (h x) \cdot 143}{2}$ and $h$ .

$\frac{145 \sin (h x) h}{2} + \frac{\cos (h x) \cdot 143 h}{2}$

Step 2.3.8

Move $143$ to the left of $\cos (h x)$ .

$\frac{145 \sin (h x) h}{2} + \frac{143 \cos (h x) h}{2}$

Step 2.4

Reorder factors in $\frac{145 \sin (h x) h}{2} + \frac{143 \cos (h x) h}{2}$ .

$\frac{145 h \sin (h x)}{2} + \frac{143 h \cos (h x)}{2}$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $\frac{145 h \sin (h x)}{2} + \frac{143 h \cos (h x)}{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{145 h \sin (h x)}{2}] + \frac{d}{d x} [\frac{143 h \cos (h x)}{2}]$ .

$f'' (x) = \frac{d}{d x} (\frac{145 h \sin (h x)}{2}) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{145 h \sin (h x)}{2}]$ .

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Step 3.2.1

Since $\frac{145 h}{2}$ is constant with respect to $x$ , the derivative of $\frac{145 h \sin (h x)}{2}$ with respect to $x$ is $\frac{145 h}{2} \frac{d}{d x} [\sin (h x)]$ .

$f'' (x) = \frac{145 h}{2} \cdot \frac{d}{d x} (\sin (h x)) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

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 $h x$ .

$f'' (x) = \frac{145 h}{2} \cdot (\frac{d}{d u (1)} (\sin (u_{1})) \frac{d}{d x} (h x)) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.2.2

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

$f'' (x) = \frac{145 h}{2} \cdot (\cos (u_{1}) \frac{d}{d x} (h x)) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.2.3

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

$f'' (x) = \frac{145 h}{2} \cdot (\cos (h x) \frac{d}{d x} (h x)) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.3

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

$f'' (x) = \frac{145 h}{2} \cdot (\cos (h x) (h \frac{d}{d x} (x))) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.4

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

$f'' (x) = \frac{145 h}{2} \cdot (\cos (h x) (h \cdot 1)) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.5

Multiply $h$ by $1$ .

$f'' (x) = \frac{145 h}{2} \cdot (\cos (h x) h) + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.6

Combine $\cos (h x)$ and $\frac{145 h}{2}$ .

$f'' (x) = \frac{\cos (h x) (145 h)}{2} \cdot h + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.7

Combine $\frac{\cos (h x) (145 h)}{2}$ and $h$ .

$f'' (x) = \frac{\cos (h x) (145 h) h}{2} + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.8

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

$f'' (x) = \frac{\cos (h x) (145 (h \cdot h))}{2} + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.9

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

$f'' (x) = \frac{\cos (h x) (145 (h \cdot h))}{2} + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.10

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

$f'' (x) = \frac{\cos (h x) (145 h^{1 + 1})}{2} + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.2.11

Add $1$ and $1$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{d}{d x} (\frac{143 h \cos (h x)}{2})$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{143 h \cos (h x)}{2}]$ .

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Step 3.3.1

Since $\frac{143 h}{2}$ is constant with respect to $x$ , the derivative of $\frac{143 h \cos (h x)}{2}$ with respect to $x$ is $\frac{143 h}{2} \frac{d}{d x} [\cos (h x)]$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot \frac{d}{d x} (\cos (h x))$

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 $h x$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (\frac{d}{d u (2)} (\cos (u_{2})) \frac{d}{d x} (h x))$

Step 3.3.2.2

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (- \sin (u_{2}) \frac{d}{d x} (h x))$

Step 3.3.2.3

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (- \sin (h x) \frac{d}{d x} (h x))$

Step 3.3.3

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (- \sin (h x) (h \frac{d}{d x} (x)))$

Step 3.3.4

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (- \sin (h x) (h \cdot 1))$

Step 3.3.5

Multiply $h$ by $1$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} + \frac{143 h}{2} \cdot (- \sin (h x) h)$

Step 3.3.6

Combine $\frac{143 h}{2}$ and $\sin (h x)$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{143 h \sin (h x)}{2} \cdot h$

Step 3.3.7

Combine $h$ and $\frac{143 h \sin (h x)}{2}$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{h (143 h \sin (h x))}{2}$

Step 3.3.8

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{143 (h \cdot h) \sin (h x)}{2}$

Step 3.3.9

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{143 (h \cdot h) \sin (h x)}{2}$

Step 3.3.10

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

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{143 h^{1 + 1} \sin (h x)}{2}$

Step 3.3.11

Add $1$ and $1$ .

$f'' (x) = \frac{\cos (h x) (145 h^{2})}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Simplify the numerator.

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Step 3.4.1.1.1

Rewrite.

$f'' (x) = \frac{0 + 0 + \cos (h x) (145 h^{2})}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 3.4.1.1.2

Add $0$ and $0$ .

$f'' (x) = \frac{0 + \cos (h x) (145 h^{2})}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 3.4.1.1.3

Remove unnecessary parentheses.

$f'' (x) = \frac{\cos (h x) \cdot (145 h^{2})}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 3.4.1.2

Move $145$ to the left of $\cos (h x)$ .

$f'' (x) = \frac{145 \cos (h x) h^{2}}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 3.4.2

Reorder factors in $\frac{145 \cos (h x) h^{2}}{2} - \frac{143 h^{2} \sin (h x)}{2}$ .

$f'' (x) = \frac{145 h^{2} \cos (h x)}{2} - \frac{143 h^{2} \sin (h x)}{2}$

Step 4

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

$\frac{145 h \sin (h x)}{2} + \frac{143 h \cos (h x)}{2} = 0$

Step 5

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

$\frac{\frac{145 h \sin (h x)}{2}}{\cos (h x)} + \frac{\frac{143 h \cos (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6

Separate fractions.

$\frac{\frac{145 h \sin (h x)}{2}}{1} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 7

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

$\frac{\frac{145 h \sin (h x)}{2}}{1} \cdot \sec (h x) + \frac{\frac{143 h \cos (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 8

Divide $\frac{145 h \sin (h x)}{2}$ by $1$ .

$\frac{145 h \sin (h x)}{2} \cdot \sec (h x) + \frac{\frac{143 h \cos (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 9

Combine $\frac{145 h \sin (h x)}{2}$ and $\sec (h x)$ .

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{\frac{143 h \cos (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 10

Separate fractions.

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{\frac{143 h \cos (h x)}{2}}{1} \cdot \frac{1}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 11

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

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{\frac{143 h \cos (h x)}{2}}{1} \cdot \sec (h x) = \frac{0}{\cos (h x)}$

Step 12

Divide $\frac{143 h \cos (h x)}{2}$ by $1$ .

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x)}{2} \cdot \sec (h x) = \frac{0}{\cos (h x)}$

Step 13

Combine $\frac{143 h \cos (h x)}{2}$ and $\sec (h x)$ .

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} = \frac{0}{\cos (h x)}$

Step 14

Separate fractions.

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} = \frac{0}{1} \cdot \frac{1}{\cos (h x)}$

Step 15

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

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} = \frac{0}{1} \cdot \sec (h x)$

Step 16

Divide $0$ by $1$ .

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} = 0 \sec (h x)$

Step 17

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

$\frac{145 h \sin (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} = 0$

Step 18

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

$\frac{\frac{145 h \sin (h x) \sec (h x)}{2}}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 19

Multiply the numerator by the reciprocal of the denominator.

$\frac{145 h \sin (h x) \sec (h x)}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 20

Simplify the numerator.

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Step 20.1

Rewrite $\sec (h x)$ in terms of sines and cosines.

$\frac{145 h \sin (h x) (\frac{1}{\cos (h x)})}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 20.2

Combine exponents.

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Step 20.2.1

Combine $145$ and $\frac{1}{\cos (h x)}$ .

$\frac{h \sin (h x) (\frac{145}{\cos (h x)})}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 20.2.2

Combine $\frac{145}{\cos (h x)}$ and $h$ .

$\frac{\frac{145 h}{\cos (h x)} \cdot \sin (h x)}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 20.2.3

Combine $\frac{145 h}{\cos (h x)}$ and $\sin (h x)$ .

$\frac{\frac{145 h \sin (h x)}{\cos (h x)}}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 21

Multiply the numerator by the reciprocal of the denominator.

$\frac{145 h \sin (h x)}{\cos (h x)} \cdot \frac{1}{2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 22

Multiply $\frac{145 h \sin (h x)}{\cos (h x)}$ by $\frac{1}{2}$ .

$\frac{145 h \sin (h x)}{\cos (h x) \cdot 2} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 23

Move $2$ to the left of $\cos (h x)$ .

$\frac{145 h \sin (h x)}{2 \cdot \cos (h x)} \cdot \frac{1}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 24

Multiply $\frac{145 h \sin (h x)}{2 \cos (h x)} \cdot \frac{1}{\cos (h x)}$ .

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Step 24.1

Multiply $\frac{145 h \sin (h x)}{2 \cos (h x)}$ by $\frac{1}{\cos (h x)}$ .

$\frac{145 h \sin (h x)}{2 \cos (h x) \cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 24.2

Raise $\cos (h x)$ to the power of $1$ .

$\frac{145 h \sin (h x)}{2 (\cos (h x) \cos (h x))} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 24.3

Raise $\cos (h x)$ to the power of $1$ .

$\frac{145 h \sin (h x)}{2 (\cos (h x) \cos (h x))} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 24.4

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

$\frac{145 h \sin (h x)}{2 {\cos (h x)}^{1 + 1}} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 24.5

Add $1$ and $1$ .

$\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 25

Factor $\cos (h x)$ out of $\cos^{2} (h x)$ .

$\frac{145 h \sin (h x)}{2 (\cos (h x) \cos (h x))} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 26

Separate fractions.

$\frac{145 h}{2 (\cos (h x))} \cdot \frac{\sin (h x)}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 27

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

$\frac{145 h}{2 (\cos (h x))} \cdot \tan (h x) + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 28

Combine $\frac{145 h}{2 \cos (h x)}$ and $\tan (h x)$ .

$\frac{145 h \tan (h x)}{2 \cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 29

Separate fractions.

$\frac{145 h}{2} \cdot \frac{\tan (h x)}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 30

Rewrite $\tan (h x)$ in terms of sines and cosines.

$\frac{145 h}{2} \cdot \frac{\frac{\sin (h x)}{\cos (h x)}}{\cos (h x)} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 31

Rewrite $\frac{\frac{\sin (h x)}{\cos (h x)}}{\cos (h x)}$ as a product.

$\frac{145 h}{2} \cdot (\frac{\sin (h x)}{\cos (h x)} \cdot \frac{1}{\cos (h x)}) + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 32

Simplify.

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Step 32.1

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

$\frac{145 h}{2} \cdot (\tan (h x) (\frac{1}{\cos (h x)})) + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 32.2

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

$\frac{145 h}{2} \cdot (\tan (h x) \sec (h x)) + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 33

Multiply $\frac{145 h}{2} (\tan (h x) \sec (h x))$ .

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Step 33.1

Combine $\tan (h x)$ and $\frac{145 h}{2}$ .

$\frac{\tan (h x) (145 h)}{2} \cdot \sec (h x) + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 33.2

Combine $\frac{\tan (h x) (145 h)}{2}$ and $\sec (h x)$ .

$\frac{\tan (h x) (145 h) \sec (h x)}{2} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 34

Remove unnecessary parentheses.

$\frac{\tan (h x) \cdot (145 h \sec (h x))}{2} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 35

Simplify the expression.

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Step 35.1

Move $145$ to the left of $\tan (h x)$ .

$\frac{145 \cdot (\tan (h x) h \sec (h x))}{2} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 35.2

Reorder factors in $\frac{145 \tan (h x) h \sec (h x)}{2}$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{\frac{143 h \cos (h x) \sec (h x)}{2}}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 36

Multiply the numerator by the reciprocal of the denominator.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \cos (h x) \sec (h x)}{2} \cdot \frac{1}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 37

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

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Step 37.1

Factor $\cos (h x)$ out of $143 h \cos (h x) \sec (h x)$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{\cos (h x) (143 h \sec (h x))}{2} \cdot \frac{1}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 37.2

Cancel the common factor.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{\cos (h x) (143 h \sec (h x))}{2} \cdot \frac{1}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 37.3

Rewrite the expression.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = \frac{0}{\cos (h x)}$

Step 38

Simplify the numerator.

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Step 38.1

Rewrite $\sec (h x)$ in terms of sines and cosines.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h (\frac{1}{\cos (h x)})}{2} = \frac{0}{\cos (h x)}$

Step 38.2

Combine exponents.

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Step 38.2.1

Combine $143$ and $\frac{1}{\cos (h x)}$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{h (\frac{143}{\cos (h x)})}{2} = \frac{0}{\cos (h x)}$

Step 38.2.2

Combine $h$ and $\frac{143}{\cos (h x)}$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{\frac{h \cdot 143}{\cos (h x)}}{2} = \frac{0}{\cos (h x)}$

Step 38.3

Move $143$ to the left of $h$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{\frac{143 h}{\cos (h x)}}{2} = \frac{0}{\cos (h x)}$

Step 39

Multiply the numerator by the reciprocal of the denominator.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h}{\cos (h x)} \cdot \frac{1}{2} = \frac{0}{\cos (h x)}$

Step 40

Multiply $\frac{143 h}{\cos (h x)}$ by $\frac{1}{2}$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h}{\cos (h x) \cdot 2} = \frac{0}{\cos (h x)}$

Step 41

Move $2$ to the left of $\cos (h x)$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h}{2 \cos (h x)} = \frac{0}{\cos (h x)}$

Step 42

Replace $\cos (h x)$ with an equivalent expression in the numerator.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \cdot \sec (h x)}{2} = \frac{0}{\cos (h x)}$

Step 43

Multiply $143 h$ by $\sec (h x)$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = \frac{0}{\cos (h x)}$

Step 44

Separate fractions.

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = \frac{0}{1} \cdot \frac{1}{\cos (h x)}$

Step 45

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

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = \frac{0}{1} \cdot \sec (h x)$

Step 46

Divide $0$ by $1$ .

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = 0 \sec (h x)$

Step 47

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

$\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2} = 0$

Step 48

Multiply both sides by $\cos (h x)$ .

$(\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49

Simplify.

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Step 49.1

Simplify the left side.

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Step 49.1.1

Simplify $(\frac{145 h \tan (h x) \sec (h x)}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x)$ .

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Step 49.1.1.1

Simplify each term.

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Step 49.1.1.1.1

Simplify the numerator.

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Step 49.1.1.1.1.1

Rewrite $\tan (h x)$ in terms of sines and cosines.

$(\frac{145 h \frac{\sin (h x)}{\cos (h x)} \sec (h x)}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.2

Rewrite $\sec (h x)$ in terms of sines and cosines.

$(\frac{145 h \frac{\sin (h x)}{\cos (h x)} \frac{1}{\cos (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3

Combine exponents.

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Step 49.1.1.1.1.3.1

Combine $145$ and $\frac{\sin (h x)}{\cos (h x)}$ .

$(\frac{h \frac{145 \sin (h x)}{\cos (h x)} \frac{1}{\cos (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.2

Combine $h$ and $\frac{145 \sin (h x)}{\cos (h x)}$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos (h x)} \cdot \frac{1}{\cos (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.3

Multiply $\frac{h (145 \sin (h x))}{\cos (h x)}$ by $\frac{1}{\cos (h x)}$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos (h x) \cos (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.4

Raise $\cos (h x)$ to the power of $1$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos^{1} (h x) \cos (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.5

Raise $\cos (h x)$ to the power of $1$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos^{1} (h x) \cos^{1} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.6

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

$(\frac{\frac{h (145 \sin (h x))}{{\cos (h x)}^{1 + 1}}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.3.7

Add $1$ and $1$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos^{2} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.4

Rewrite.

$(\frac{\frac{h (145 \sin (h x)) \cdot 1}{\cos^{2} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.5

Multiply $h (145 \sin (h x))$ by $1$ .

$(\frac{\frac{h (145 \sin (h x))}{\cos^{2} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.6

Remove unnecessary parentheses.

$(\frac{\frac{h \cdot 145 \sin (h x)}{\cos^{2} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.1.7

Move $145$ to the left of $h$ .

$(\frac{\frac{145 h \sin (h x)}{\cos^{2} (h x)}}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.2

Multiply the numerator by the reciprocal of the denominator.

$(\frac{145 h \sin (h x)}{\cos^{2} (h x)} \cdot \frac{1}{2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.3

Combine.

$(\frac{145 h \sin (h x) \cdot 1}{\cos^{2} (h x) \cdot 2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.4

Multiply $145$ by $1$ .

$(\frac{145 h \sin (h x)}{\cos^{2} (h x) \cdot 2} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.5

Move $2$ to the left of $\cos^{2} (h x)$ .

$(\frac{145 h \sin (h x)}{2 \cdot \cos^{2} (h x)} + \frac{143 h \sec (h x)}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.6

Simplify the numerator.

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Step 49.1.1.1.6.1

Rewrite $\sec (h x)$ in terms of sines and cosines.

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{143 h \frac{1}{\cos (h x)}}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.6.2

Combine exponents.

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Step 49.1.1.1.6.2.1

Combine $143$ and $\frac{1}{\cos (h x)}$ .

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{h \frac{143}{\cos (h x)}}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.6.2.2

Combine $h$ and $\frac{143}{\cos (h x)}$ .

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{\frac{h \cdot 143}{\cos (h x)}}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.6.3

Move $143$ to the left of $h$ .

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{\frac{143 h}{\cos (h x)}}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.7

Multiply the numerator by the reciprocal of the denominator.

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{143 h}{\cos (h x)} \cdot \frac{1}{2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.8

Multiply $\frac{143 h}{\cos (h x)}$ by $\frac{1}{2}$ .

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{143 h}{\cos (h x) \cdot 2}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.1.9

Move $2$ to the left of $\cos (h x)$ .

$(\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} + \frac{143 h}{2 \cos (h x)}) \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2

Simplify terms.

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Step 49.1.1.2.1

Apply the distributive property.

$\frac{145 h \sin (h x)}{2 \cos^{2} (h x)} \cos (h x) + \frac{143 h}{2 \cos (h x)} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.2

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

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Step 49.1.1.2.2.1

Factor $\cos (h x)$ out of $2 \cos^{2} (h x)$ .

$\frac{145 h \sin (h x)}{\cos (h x) (2 \cos (h x))} \cos (h x) + \frac{143 h}{2 \cos (h x)} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.2.2

Cancel the common factor.

$\frac{145 h \sin (h x)}{\cos (h x) (2 \cos (h x))} \cos (h x) + \frac{143 h}{2 \cos (h x)} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.2.3

Rewrite the expression.

$\frac{145 h \sin (h x)}{2 \cos (h x)} + \frac{143 h}{2 \cos (h x)} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.3

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

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Step 49.1.1.2.3.1

Factor $\cos (h x)$ out of $2 \cos (h x)$ .

$\frac{145 h \sin (h x)}{2 \cos (h x)} + \frac{143 h}{\cos (h x) \cdot 2} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.3.2

Cancel the common factor.

$\frac{145 h \sin (h x)}{2 \cos (h x)} + \frac{143 h}{\cos (h x) \cdot 2} \cos (h x) = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.2.3.3

Rewrite the expression.

$\frac{145 h \sin (h x)}{2 \cos (h x)} + \frac{143 h}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.3

Simplify each term.

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Step 49.1.1.3.1

Separate fractions.

$\frac{145 h}{2} \cdot \frac{\sin (h x)}{\cos (h x)} + \frac{143 h}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.3.2

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

$\frac{145 h}{2} \tan (h x) + \frac{143 h}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.3.3

Combine $\frac{145 h}{2}$ and $\tan (h x)$ .

$\frac{145 h \tan (h x)}{2} + \frac{143 h}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.1.1.4

Reorder $\frac{145 h \tan (h x)}{2}$ and $\frac{143 h}{2}$ .

$\frac{143 h}{2} + \frac{145 h \tan (h x)}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.2

Simplify the right side.

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Step 49.2.1

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

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Step 49.2.1.1

Cancel the common factor.

$\frac{143 h}{2} + \frac{145 h \tan (h x)}{2} = \frac{0}{\cos (h x)} \cos (h x)$

Step 49.2.1.2

Rewrite the expression.

$\frac{143 h}{2} + \frac{145 h \tan (h x)}{2} = 0$

Step 50

Solve for $x$ .

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Step 50.1

Subtract $\frac{143 h}{2}$ from both sides of the equation.

$\frac{145 h \tan (h x)}{2} = - \frac{143 h}{2}$

Step 50.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$145 h \tan (h x) = - 143 h$

Step 50.3

Divide each term in $145 h \tan (h x) = - 143 h$ by $145 h$ and simplify.

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Step 50.3.1

Divide each term in $145 h \tan (h x) = - 143 h$ by $145 h$ .

$\frac{145 h \tan (h x)}{145 h} = \frac{- 143 h}{145 h}$

Step 50.3.2

Simplify the left side.

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Step 50.3.2.1

Cancel the common factor of $145$ .

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Step 50.3.2.1.1

Cancel the common factor.

$\frac{145 h \tan (h x)}{145 h} = \frac{- 143 h}{145 h}$

Step 50.3.2.1.2

Rewrite the expression.

$\frac{h \tan (h x)}{h} = \frac{- 143 h}{145 h}$

Step 50.3.2.2

Cancel the common factor of $h$ .

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Step 50.3.2.2.1

Cancel the common factor.

$\frac{h \tan (h x)}{h} = \frac{- 143 h}{145 h}$

Step 50.3.2.2.2

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

$\tan (h x) = \frac{- 143 h}{145 h}$

Step 50.3.3

Simplify the right side.

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Step 50.3.3.1

Cancel the common factor of $h$ .

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Step 50.3.3.1.1

Cancel the common factor.

$\tan (h x) = \frac{- 143 h}{145 h}$

Step 50.3.3.1.2

Rewrite the expression.

$\tan (h x) = \frac{- 143}{145}$

Step 50.3.3.2

Move the negative in front of the fraction.

$\tan (h x) = - \frac{143}{145}$

Step 50.4

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

$h x = \arctan (- \frac{143}{145})$

Step 50.5

Simplify the right side.

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Step 50.5.1

Evaluate $\arctan (- \frac{143}{145})$ .

$h x = - 0.77845383$

Step 50.6

Divide each term in $h x = - 0.77845383$ by $h$ and simplify.

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Step 50.6.1

Divide each term in $h x = - 0.77845383$ by $h$ .

$\frac{h x}{h} = \frac{- 0.77845383}{h}$

Step 50.6.2

Simplify the left side.

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Step 50.6.2.1

Cancel the common factor of $h$ .

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Step 50.6.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{- 0.77845383}{h}$

Step 50.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.77845383}{h}$

Step 50.6.3

Simplify the right side.

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Step 50.6.3.1

Move the negative in front of the fraction.

$x = - \frac{0.77845383}{h}$

Step 50.7

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.

$h x = - 0.77845383 - (3.14159265)$

Step 50.8

Add $2 π$ to $- 0.77845383 - (3.14159265)$ .

$h x = - 0.77845383 - (3.14159265) + 2 π$

Step 50.9

The resulting angle of $2.36313882$ is positive and coterminal with $- 0.77845383 - (3.14159265)$ .

$h x = 2.36313882$

Step 50.10

Divide each term in $h x = 2.36313882$ by $h$ and simplify.

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Step 50.10.1

Divide each term in $h x = 2.36313882$ by $h$ .

$\frac{h x}{h} = \frac{2.36313882}{h}$

Step 50.10.2

Simplify the left side.

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Step 50.10.2.1

Cancel the common factor of $h$ .

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Step 50.10.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{2.36313882}{h}$

Step 50.10.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.36313882}{h}$

Step 51

Evaluate the second derivative at $x = \frac{2.36313882}{h}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{145 h^{2} \cos (h (\frac{2.36313882}{h}))}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52

Evaluate the second derivative.

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Step 52.1

Simplify each term.

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Step 52.1.1

Simplify the numerator.

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Step 52.1.1.1

Combine $h$ and $\frac{2.36313882}{h}$ .

$\frac{145 h^{2} \cos (\frac{h \cdot 2.36313882}{h})}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.1.2

Cancel the common factor of $h$ .

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Step 52.1.1.2.1

Cancel the common factor.

$\frac{145 h^{2} \cos (\frac{h \cdot 2.36313882}{h})}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.1.2.2

Divide $2.36313882$ by $1$ .

$\frac{145 h^{2} \cos (2.36313882)}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.1.3

Evaluate $\cos (2.36313882)$ .

$\frac{145 h^{2} \cdot - 0.71200007}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.1.4

Multiply $- 0.71200007$ by $145$ .

$\frac{- 103.24001115 h^{2}}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.2

Move the negative in front of the fraction.

$- \frac{103.24001115 h^{2}}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.3

Factor $103.24001115$ out of $103.24001115 h^{2}$ .

$- \frac{103.24001115 (h^{2})}{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.4

Factor $2$ out of $2$ .

$- \frac{103.24001115 (h^{2})}{2 (1)} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.5

Separate fractions.

$- (\frac{103.24001115}{2} \cdot \frac{h^{2}}{1}) - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.6

Divide $103.24001115$ by $2$ .

$- (51.62000557 \frac{h^{2}}{1}) - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.7

Divide $h^{2}$ by $1$ .

$- (51.62000557 h^{2}) - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.8

Multiply $51.62000557$ by $- 1$ .

$- 51.62000557 h^{2} - \frac{143 h^{2} \sin (h (\frac{2.36313882}{h}))}{2}$

Step 52.1.9

Simplify the numerator.

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Step 52.1.9.1

Cancel the common factor of $h$ .

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Step 52.1.9.1.1

Cancel the common factor.

$- 51.62000557 h^{2} - \frac{143 h^{2} \sin (h \frac{2.36313882}{h})}{2}$

Step 52.1.9.1.2

Rewrite the expression.

$- 51.62000557 h^{2} - \frac{143 h^{2} \sin (2.36313882)}{2}$

Step 52.1.9.2

Evaluate $\sin (2.36313882)$ .

$- 51.62000557 h^{2} - \frac{143 h^{2} \cdot 0.70217938}{2}$

Step 52.1.9.3

Multiply $0.70217938$ by $143$ .

$- 51.62000557 h^{2} - \frac{100.41165222 h^{2}}{2}$

Step 52.1.10

Factor $100.41165222$ out of $100.41165222 h^{2}$ .

$- 51.62000557 h^{2} - \frac{100.41165222 (h^{2})}{2}$

Step 52.1.11

Factor $2$ out of $2$ .

$- 51.62000557 h^{2} - \frac{100.41165222 (h^{2})}{2 (1)}$

Step 52.1.12

Separate fractions.

$- 51.62000557 h^{2} - (\frac{100.41165222}{2} \cdot \frac{h^{2}}{1})$

Step 52.1.13

Divide $100.41165222$ by $2$ .

$- 51.62000557 h^{2} - (50.20582611 \frac{h^{2}}{1})$

Step 52.1.14

Divide $h^{2}$ by $1$ .

$- 51.62000557 h^{2} - (50.20582611 h^{2})$

Step 52.1.15

Multiply $50.20582611$ by $- 1$ .

$- 51.62000557 h^{2} - 50.20582611 h^{2}$

Step 52.2

Subtract $50.20582611 h^{2}$ from $- 51.62000557 h^{2}$ .

$- 101.82583169 h^{2}$

Step 53

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

No Local Extrema

Step 54