Calculus Examples

$y = \frac{5 + \csc (x)}{9 - \csc (x)}$ , $(\frac{π}{6}, 1)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{6}$ and $y = 1$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 5 + \csc (x)$ and $g (x) = 9 - \csc (x)$ .

$\frac{(9 - \csc (x)) \frac{d}{d x} [5 + \csc (x)] - (5 + \csc (x)) \frac{d}{d x} [9 - \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

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

$\frac{(9 - \csc (x)) (\frac{d}{d x} [5] + \frac{d}{d x} [\csc (x)]) - (5 + \csc (x)) \frac{d}{d x} [9 - \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.2.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{(9 - \csc (x)) (0 + \frac{d}{d x} [\csc (x)]) - (5 + \csc (x)) \frac{d}{d x} [9 - \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [\csc (x)]$ .

$\frac{(9 - \csc (x)) \frac{d}{d x} [\csc (x)] - (5 + \csc (x)) \frac{d}{d x} [9 - \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.3

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

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) - (5 + \csc (x)) \frac{d}{d x} [9 - \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.4

Differentiate.

Tap for more steps...

Step 1.4.1

By the Sum Rule, the derivative of $9 - \csc (x)$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- \csc (x)]$ .

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) - (5 + \csc (x)) (\frac{d}{d x} [9] + \frac{d}{d x} [- \csc (x)])}{{(9 - \csc (x))}^{2}}$

Step 1.4.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) - (5 + \csc (x)) (0 + \frac{d}{d x} [- \csc (x)])}{{(9 - \csc (x))}^{2}}$

Step 1.4.3

Add $0$ and $\frac{d}{d x} [- \csc (x)]$ .

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) - (5 + \csc (x)) \frac{d}{d x} [- \csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.4.4

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

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) - (5 + \csc (x)) (- \frac{d}{d x} [\csc (x)])}{{(9 - \csc (x))}^{2}}$

Step 1.4.5

Multiply.

Tap for more steps...

Step 1.4.5.1

Multiply $- 1$ by $- 1$ .

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) + 1 (5 + \csc (x)) \frac{d}{d x} [\csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.4.5.2

Multiply $5 + \csc (x)$ by $1$ .

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) + (5 + \csc (x)) \frac{d}{d x} [\csc (x)]}{{(9 - \csc (x))}^{2}}$

Step 1.5

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

$\frac{(9 - \csc (x)) (- \csc (x) \cot (x)) + (5 + \csc (x)) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Apply the distributive property.

$\frac{9 (- \csc (x) \cot (x)) - \csc (x) (- \csc (x) \cot (x)) + (5 + \csc (x)) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.2

Apply the distributive property.

$\frac{9 (- \csc (x) \cot (x)) - \csc (x) (- \csc (x) \cot (x)) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3

Simplify the numerator.

Tap for more steps...

Step 1.6.3.1

Simplify each term.

Tap for more steps...

Step 1.6.3.1.1

Multiply $- 1$ by $9$ .

$\frac{- 9 (\csc (x) \cot (x)) - \csc (x) (- \csc (x) \cot (x)) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2

Multiply $- \csc (x) (- \csc (x) \cot (x))$ .

Tap for more steps...

Step 1.6.3.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- 9 \csc (x) \cot (x) + 1 \csc (x) (\csc (x) \cot (x)) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2.2

Multiply $\csc (x)$ by $1$ .

$\frac{- 9 \csc (x) \cot (x) + \csc (x) (\csc (x) \cot (x)) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2.3

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

$\frac{- 9 \csc (x) \cot (x) + \csc^{1} (x) \csc (x) \cot (x) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2.4

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

$\frac{- 9 \csc (x) \cot (x) + \csc^{1} (x) \csc^{1} (x) \cot (x) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2.5

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

$\frac{- 9 \csc (x) \cot (x) + {\csc (x)}^{1 + 1} \cot (x) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.2.6

Add $1$ and $1$ .

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) + 5 (- \csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.3

Multiply $- 1$ by $5$ .

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 (\csc (x) \cot (x)) + \csc (x) (- \csc (x) \cot (x))}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - \csc (x) \csc (x) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.5

Multiply $- \csc (x) \csc (x)$ .

Tap for more steps...

Step 1.6.3.1.5.1

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

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - (\csc^{1} (x) \csc (x)) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.5.2

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

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - (\csc^{1} (x) \csc^{1} (x)) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.5.3

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

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - {\csc (x)}^{1 + 1} \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.1.5.4

Add $1$ and $1$ .

$\frac{- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - \csc^{2} (x) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.2

Combine the opposite terms in $- 9 \csc (x) \cot (x) + \csc^{2} (x) \cot (x) - 5 \csc (x) \cot (x) - \csc^{2} (x) \cot (x)$ .

Tap for more steps...

Step 1.6.3.2.1

Subtract $\csc^{2} (x) \cot (x)$ from $\csc^{2} (x) \cot (x)$ .

$\frac{- 9 \csc (x) \cot (x) - 5 \csc (x) \cot (x) + 0}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.2.2

Add $- 9 \csc (x) \cot (x) - 5 \csc (x) \cot (x)$ and $0$ .

$\frac{- 9 \csc (x) \cot (x) - 5 \csc (x) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.3.3

Subtract $5 \csc (x) \cot (x)$ from $- 9 \csc (x) \cot (x)$ .

$\frac{- 14 \csc (x) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.6.4

Move the negative in front of the fraction.

$- \frac{14 \csc (x) \cot (x)}{{(9 - \csc (x))}^{2}}$

Step 1.7

Evaluate the derivative at $x = \frac{π}{6}$ .

$- \frac{14 \csc (\frac{π}{6}) \cot (\frac{π}{6})}{{(9 - \csc (\frac{π}{6}))}^{2}}$

Step 1.8

Simplify.

Tap for more steps...

Step 1.8.1

Simplify the numerator.

Tap for more steps...

Step 1.8.1.1

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$- \frac{14 \cdot 2 \cot (\frac{π}{6})}{{(9 - \csc (\frac{π}{6}))}^{2}}$

Step 1.8.1.2

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$- \frac{14 \cdot 2 \sqrt{3}}{{(9 - \csc (\frac{π}{6}))}^{2}}$

Step 1.8.1.3

Multiply $14$ by $2$ .

$- \frac{28 \sqrt{3}}{{(9 - \csc (\frac{π}{6}))}^{2}}$

Step 1.8.2

Simplify the denominator.

Tap for more steps...

Step 1.8.2.1

The exact value of $\csc (\frac{π}{6})$ is $2$ .

$- \frac{28 \sqrt{3}}{{(9 - 1 \cdot 2)}^{2}}$

Step 1.8.2.2

Multiply $- 1$ by $2$ .

$- \frac{28 \sqrt{3}}{{(9 - 2)}^{2}}$

Step 1.8.2.3

Subtract $2$ from $9$ .

$- \frac{28 \sqrt{3}}{7^{2}}$

Step 1.8.2.4

Raise $7$ to the power of $2$ .

$- \frac{28 \sqrt{3}}{49}$

Step 1.8.3

Cancel the common factor of $28$ and $49$ .

Tap for more steps...

Step 1.8.3.1

Factor $7$ out of $28 \sqrt{3}$ .

$- \frac{7 (4 \sqrt{3})}{49}$

Step 1.8.3.2

Cancel the common factors.

Tap for more steps...

Step 1.8.3.2.1

Factor $7$ out of $49$ .

$- \frac{7 (4 \sqrt{3})}{7 (7)}$

Step 1.8.3.2.2

Cancel the common factor.

$- \frac{7 (4 \sqrt{3})}{7 \cdot 7}$

Step 1.8.3.2.3

Rewrite the expression.

$- \frac{4 \sqrt{3}}{7}$

Step 2

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

Tap for more steps...

Step 2.1

Use the slope $- \frac{4 \sqrt{3}}{7}$ and a given point $(\frac{π}{6}, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{4 \sqrt{3}}{7} \cdot (x - (\frac{π}{6}))$

Step 2.2

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

$y - 1 = - \frac{4 \sqrt{3}}{7} \cdot (x - \frac{π}{6})$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $- \frac{4 \sqrt{3}}{7} \cdot (x - \frac{π}{6})$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{4 \sqrt{3}}{7} \cdot (x - \frac{π}{6})$

Step 2.3.1.2

Simplify terms.

Tap for more steps...

Step 2.3.1.2.1

Apply the distributive property.

$y - 1 = - \frac{4 \sqrt{3}}{7} x - \frac{4 \sqrt{3}}{7} (- \frac{π}{6})$

Step 2.3.1.2.2

Combine $x$ and $\frac{4 \sqrt{3}}{7}$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} - \frac{4 \sqrt{3}}{7} (- \frac{π}{6})$

Step 2.3.1.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.2.3.1

Move the leading negative in $- \frac{4 \sqrt{3}}{7}$ into the numerator.

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{- 4 \sqrt{3}}{7} (- \frac{π}{6})$

Step 2.3.1.2.3.2

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{- 4 \sqrt{3}}{7} \cdot \frac{- π}{6}$

Step 2.3.1.2.3.3

Factor $2$ out of $- 4 \sqrt{3}$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{2 (- 2 \sqrt{3})}{7} \cdot \frac{- π}{6}$

Step 2.3.1.2.3.4

Factor $2$ out of $6$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{2 (- 2 \sqrt{3})}{7} \cdot \frac{- π}{2 (3)}$

Step 2.3.1.2.3.5

Cancel the common factor.

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{2 (- 2 \sqrt{3})}{7} \cdot \frac{- π}{2 \cdot 3}$

Step 2.3.1.2.3.6

Rewrite the expression.

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{- 2 \sqrt{3}}{7} \cdot \frac{- π}{3}$

Step 2.3.1.2.4

Multiply $\frac{- 2 \sqrt{3}}{7}$ by $\frac{- π}{3}$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{- 2 \sqrt{3} (- π)}{7 \cdot 3}$

Step 2.3.1.2.5

Multiply.

Tap for more steps...

Step 2.3.1.2.5.1

Multiply $- 1$ by $- 2$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{2 \sqrt{3} π}{7 \cdot 3}$

Step 2.3.1.2.5.2

Multiply $7$ by $3$ .

$y - 1 = - \frac{x (4 \sqrt{3})}{7} + \frac{2 \sqrt{3} π}{21}$

Step 2.3.1.3

Move $4$ to the left of $x$ .

$y - 1 = - \frac{4 x \sqrt{3}}{7} + \frac{2 \sqrt{3} π}{21}$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = - \frac{4 x \sqrt{3}}{7} + \frac{2 \sqrt{3} π}{21} + 1$

Step 2.3.3

Write in $y = m x + b$ form.

Tap for more steps...

Step 2.3.3.1

Write $1$ as a fraction with a common denominator.

$y = - \frac{4 x \sqrt{3}}{7} + \frac{2 \sqrt{3} π}{21} + \frac{21}{21}$

Step 2.3.3.2

Combine the numerators over the common denominator.

$y = - \frac{4 x \sqrt{3}}{7} + \frac{2 \sqrt{3} π + 21}{21}$

Step 2.3.3.3

Reorder terms.

$y = - (\frac{4 \sqrt{3}}{7} x) + \frac{2 \sqrt{3} π + 21}{21}$

Step 2.3.3.4

Remove parentheses.

$y = - \frac{4 \sqrt{3}}{7} x + \frac{2 \sqrt{3} π + 21}{21}$

Step 3