Calculus Examples

$f (x) = - 5 \cot (x) - \frac{5 π}{2} - 4$ at $x = - \frac{π}{4}$

Step 1

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

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Step 1.1

Substitute $- \frac{π}{4}$ in for $x$ .

$y = - 5 \cot (- \frac{π}{4}) - \frac{5 π}{2} - 4$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Remove parentheses.

$y = - 5 \cot (- \frac{π}{4}) - \frac{5 π}{2} - 4$

Step 1.2.2

Simplify $- 5 \cot (- \frac{π}{4}) - \frac{5 π}{2} - 4$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$y = - 5 \cot (\frac{7 π}{4}) - \frac{5 π}{2} - 4$

Step 1.2.2.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the fourth quadrant.

$y = - 5 (- \cot (\frac{π}{4})) - \frac{5 π}{2} - 4$

Step 1.2.2.1.3

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$y = - 5 (- 1 \cdot 1) - \frac{5 π}{2} - 4$

Step 1.2.2.1.4

Multiply $- 5 (- 1 \cdot 1)$ .

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Step 1.2.2.1.4.1

Multiply $- 1$ by $1$ .

$y = - 5 \cdot - 1 - \frac{5 π}{2} - 4$

Step 1.2.2.1.4.2

Multiply $- 5$ by $- 1$ .

$y = 5 - \frac{5 π}{2} - 4$

Step 1.2.2.2

Subtract $4$ from $5$ .

$y = 1 - \frac{5 π}{2}$

Step 2

Find the first derivative and evaluate at $x = - \frac{π}{4}$ and $y = 1 - \frac{5 π}{2}$ to find the slope of the tangent line.

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Step 2.1

By the Sum Rule, the derivative of $- 5 \cot (x) - \frac{5 π}{2} - 4$ with respect to $x$ is $\frac{d}{d x} [- 5 \cot (x)] + \frac{d}{d x} [- \frac{5 π}{2}] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [- 5 \cot (x)] + \frac{d}{d x} [- \frac{5 π}{2}] + \frac{d}{d x} [- 4]$

Step 2.2

Evaluate $\frac{d}{d x} [- 5 \cot (x)]$ .

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Step 2.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 \cot (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [\cot (x)]$ .

$- 5 \frac{d}{d x} [\cot (x)] + \frac{d}{d x} [- \frac{5 π}{2}] + \frac{d}{d x} [- 4]$

Step 2.2.2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- 5 (- \csc^{2} (x)) + \frac{d}{d x} [- \frac{5 π}{2}] + \frac{d}{d x} [- 4]$

Step 2.2.3

Multiply $- 1$ by $- 5$ .

$5 \csc^{2} (x) + \frac{d}{d x} [- \frac{5 π}{2}] + \frac{d}{d x} [- 4]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- \frac{5 π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5 π}{2}$ with respect to $x$ is $0$ .

$5 \csc^{2} (x) + 0 + \frac{d}{d x} [- 4]$

Step 2.3.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$5 \csc^{2} (x) + 0 + 0$

Step 2.4

Combine terms.

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Step 2.4.1

Add $5 \csc^{2} (x)$ and $0$ .

$5 \csc^{2} (x) + 0$

Step 2.4.2

Add $5 \csc^{2} (x)$ and $0$ .

$5 \csc^{2} (x)$

Step 2.5

Evaluate the derivative at $x = - \frac{π}{4}$ .

$5 \csc^{2} (- \frac{π}{4})$

Step 2.6

Simplify.

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Step 2.6.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$5 \csc^{2} (\frac{7 π}{4})$

Step 2.6.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$5 {(- \csc (\frac{π}{4}))}^{2}$

Step 2.6.3

The exact value of $\csc (\frac{π}{4})$ is $\sqrt{2}$ .

$5 {(- \sqrt{2})}^{2}$

Step 2.6.4

Simplify the expression.

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Step 2.6.4.1

Apply the product rule to $- \sqrt{2}$ .

$5 ({(- 1)}^{2} {\sqrt{2}}^{2})$

Step 2.6.4.2

Raise $- 1$ to the power of $2$ .

$5 (1 {\sqrt{2}}^{2})$

Step 2.6.4.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$5 {\sqrt{2}}^{2}$

Step 2.6.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.6.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$5 {(2^{\frac{1}{2}})}^{2}$

Step 2.6.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 2.6.5.3

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

$5 \cdot 2^{\frac{2}{2}}$

Step 2.6.5.4

Cancel the common factor of $2$ .

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Step 2.6.5.4.1

Cancel the common factor.

$5 \cdot 2^{\frac{2}{2}}$

Step 2.6.5.4.2

Rewrite the expression.

$5 \cdot 2^{1}$

Step 2.6.5.5

Evaluate the exponent.

$5 \cdot 2$

Step 2.6.6

Multiply $5$ by $2$ .

$10$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $10$ and a given point $(- \frac{π}{4}, 1 - \frac{5 π}{2})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1 - \frac{5 π}{2}) = 10 \cdot (x - (- \frac{π}{4}))$

Step 3.2

Simplify the equation and keep it in point-slope form.

$y - 1 + \frac{5 π}{2} = 10 \cdot (x + \frac{π}{4})$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $10 \cdot (x + \frac{π}{4})$ .

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Step 3.3.1.1

Rewrite.

$y - 1 + \frac{5 π}{2} = 0 + 0 + 10 \cdot (x + \frac{π}{4})$

Step 3.3.1.2

Simplify by adding zeros.

$y - 1 + \frac{5 π}{2} = 10 \cdot (x + \frac{π}{4})$

Step 3.3.1.3

Apply the distributive property.

$y - 1 + \frac{5 π}{2} = 10 x + 10 \frac{π}{4}$

Step 3.3.1.4

Cancel the common factor of $2$ .

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Step 3.3.1.4.1

Factor $2$ out of $10$ .

$y - 1 + \frac{5 π}{2} = 10 x + 2 (5) \frac{π}{4}$

Step 3.3.1.4.2

Factor $2$ out of $4$ .

$y - 1 + \frac{5 π}{2} = 10 x + 2 \cdot 5 \frac{π}{2 \cdot 2}$

Step 3.3.1.4.3

Cancel the common factor.

$y - 1 + \frac{5 π}{2} = 10 x + 2 \cdot 5 \frac{π}{2 \cdot 2}$

Step 3.3.1.4.4

Rewrite the expression.

$y - 1 + \frac{5 π}{2} = 10 x + 5 \frac{π}{2}$

Step 3.3.1.5

Combine $5$ and $\frac{π}{2}$ .

$y - 1 + \frac{5 π}{2} = 10 x + \frac{5 π}{2}$

Step 3.3.2

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

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Step 3.3.2.1

Add $1$ to both sides of the equation.

$y + \frac{5 π}{2} = 10 x + \frac{5 π}{2} + 1$

Step 3.3.2.2

Subtract $\frac{5 π}{2}$ from both sides of the equation.

$y = 10 x + \frac{5 π}{2} + 1 - \frac{5 π}{2}$

Step 3.3.2.3

Combine the opposite terms in $10 x + \frac{5 π}{2} + 1 - \frac{5 π}{2}$ .

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Step 3.3.2.3.1

Combine the numerators over the common denominator.

$y = 10 x + \frac{5 π - 5 π}{2} + 1$

Step 3.3.2.3.2

Subtract $5 π$ from $5 π$ .

$y = 10 x + \frac{0}{2} + 1$

Step 3.3.2.4

Divide $0$ by $2$ .

$y = 10 x + 0 + 1$

Step 3.3.2.5

Add $10 x$ and $0$ .

$y = 10 x + 1$

Step 4