Calculus Examples

$f (x) = \frac{20 x}{x^{2} - 5}$ ; $(5, 5)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 5$ to find the slope of the tangent line.

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Step 1.1

Since $20$ is constant with respect to $x$ , the derivative of $\frac{20 x}{x^{2} - 5}$ with respect to $x$ is $20 \frac{d}{d x} [\frac{x}{x^{2} - 5}]$ .

$20 \frac{d}{d x} [\frac{x}{x^{2} - 5}]$

Step 1.2

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 $f (x) = x$ and $g (x) = x^{2} - 5$ .

$20 \frac{(x^{2} - 5) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 5]}{{(x^{2} - 5)}^{2}}$

Step 1.3

Differentiate.

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Step 1.3.1

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

$20 \frac{(x^{2} - 5) \cdot 1 - x \frac{d}{d x} [x^{2} - 5]}{{(x^{2} - 5)}^{2}}$

Step 1.3.2

Multiply $x^{2} - 5$ by $1$ .

$20 \frac{x^{2} - 5 - x \frac{d}{d x} [x^{2} - 5]}{{(x^{2} - 5)}^{2}}$

Step 1.3.3

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

$20 \frac{x^{2} - 5 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5])}{{(x^{2} - 5)}^{2}}$

Step 1.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$20 \frac{x^{2} - 5 - x (2 x + \frac{d}{d x} [- 5])}{{(x^{2} - 5)}^{2}}$

Step 1.3.5

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

$20 \frac{x^{2} - 5 - x (2 x + 0)}{{(x^{2} - 5)}^{2}}$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $2 x$ and $0$ .

$20 \frac{x^{2} - 5 - x (2 x)}{{(x^{2} - 5)}^{2}}$

Step 1.3.6.2

Multiply $2$ by $- 1$ .

$20 \frac{x^{2} - 5 - 2 x \cdot x}{{(x^{2} - 5)}^{2}}$

Step 1.4

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

$20 \frac{x^{2} - 5 - 2 (x^{1} x)}{{(x^{2} - 5)}^{2}}$

Step 1.5

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

$20 \frac{x^{2} - 5 - 2 (x^{1} x^{1})}{{(x^{2} - 5)}^{2}}$

Step 1.6

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

$20 \frac{x^{2} - 5 - 2 x^{1 + 1}}{{(x^{2} - 5)}^{2}}$

Step 1.7

Add $1$ and $1$ .

$20 \frac{x^{2} - 5 - 2 x^{2}}{{(x^{2} - 5)}^{2}}$

Step 1.8

Subtract $2 x^{2}$ from $x^{2}$ .

$20 \frac{- x^{2} - 5}{{(x^{2} - 5)}^{2}}$

Step 1.9

Combine $20$ and $\frac{- x^{2} - 5}{{(x^{2} - 5)}^{2}}$ .

$\frac{20 (- x^{2} - 5)}{{(x^{2} - 5)}^{2}}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

$\frac{20 (- x^{2}) + 20 \cdot - 5}{{(x^{2} - 5)}^{2}}$

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply $- 1$ by $20$ .

$\frac{- 20 x^{2} + 20 \cdot - 5}{{(x^{2} - 5)}^{2}}$

Step 1.10.2.2

Multiply $20$ by $- 5$ .

$\frac{- 20 x^{2} - 100}{{(x^{2} - 5)}^{2}}$

Step 1.10.3

Factor $20$ out of $- 20 x^{2} - 100$ .

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Step 1.10.3.1

Factor $20$ out of $- 20 x^{2}$ .

$\frac{20 (- x^{2}) - 100}{{(x^{2} - 5)}^{2}}$

Step 1.10.3.2

Factor $20$ out of $- 100$ .

$\frac{20 (- x^{2}) + 20 (- 5)}{{(x^{2} - 5)}^{2}}$

Step 1.10.3.3

Factor $20$ out of $20 (- x^{2}) + 20 (- 5)$ .

$\frac{20 (- x^{2} - 5)}{{(x^{2} - 5)}^{2}}$

Step 1.10.4

Factor $- 1$ out of $- x^{2}$ .

$\frac{20 (- (x^{2}) - 5)}{{(x^{2} - 5)}^{2}}$

Step 1.10.5

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{20 (- (x^{2}) - 1 (5))}{{(x^{2} - 5)}^{2}}$

Step 1.10.6

Factor $- 1$ out of $- (x^{2}) - 1 (5)$ .

$\frac{20 (- (x^{2} + 5))}{{(x^{2} - 5)}^{2}}$

Step 1.10.7

Rewrite $- (x^{2} + 5)$ as $- 1 (x^{2} + 5)$ .

$\frac{20 (- 1 (x^{2} + 5))}{{(x^{2} - 5)}^{2}}$

Step 1.10.8

Move the negative in front of the fraction.

$- \frac{20 (x^{2} + 5)}{{(x^{2} - 5)}^{2}}$

Step 1.11

Evaluate the derivative at $x = 5$ .

$- \frac{20 ({(5)}^{2} + 5)}{{({(5)}^{2} - 5)}^{2}}$

Step 1.12

Simplify.

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Step 1.12.1

Simplify the numerator.

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Step 1.12.1.1

Raise $5$ to the power of $2$ .

$- \frac{20 (25 + 5)}{{(5^{2} - 5)}^{2}}$

Step 1.12.1.2

Add $25$ and $5$ .

$- \frac{20 \cdot 30}{{(5^{2} - 5)}^{2}}$

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

Raise $5$ to the power of $2$ .

$- \frac{20 \cdot 30}{{(25 - 5)}^{2}}$

Step 1.12.2.2

Subtract $5$ from $25$ .

$- \frac{20 \cdot 30}{20^{2}}$

Step 1.12.2.3

Raise $20$ to the power of $2$ .

$- \frac{20 \cdot 30}{400}$

Step 1.12.3

Reduce the expression by cancelling the common factors.

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Step 1.12.3.1

Multiply $20$ by $30$ .

$- \frac{600}{400}$

Step 1.12.3.2

Cancel the common factor of $600$ and $400$ .

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Step 1.12.3.2.1

Factor $200$ out of $600$ .

$- \frac{200 (3)}{400}$

Step 1.12.3.2.2

Cancel the common factors.

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Step 1.12.3.2.2.1

Factor $200$ out of $400$ .

$- \frac{200 \cdot 3}{200 \cdot 2}$

Step 1.12.3.2.2.2

Cancel the common factor.

$- \frac{200 \cdot 3}{200 \cdot 2}$

Step 1.12.3.2.2.3

Rewrite the expression.

$- \frac{3}{2}$

Step 2

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

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Step 2.1

Use the slope $- \frac{3}{2}$ and a given point $(5, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = - \frac{3}{2} \cdot (x - (5))$

Step 2.2

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

$y - 5 = - \frac{3}{2} \cdot (x - 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{3}{2} \cdot (x - 5)$ .

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Step 2.3.1.1

Rewrite.

$y - 5 = 0 + 0 - \frac{3}{2} \cdot (x - 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 5 = - \frac{3}{2} \cdot (x - 5)$

Step 2.3.1.3

Apply the distributive property.

$y - 5 = - \frac{3}{2} x - \frac{3}{2} \cdot - 5$

Step 2.3.1.4

Combine $x$ and $\frac{3}{2}$ .

$y - 5 = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot - 5$

Step 2.3.1.5

Multiply $- \frac{3}{2} \cdot - 5$ .

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Step 2.3.1.5.1

Multiply $- 5$ by $- 1$ .

$y - 5 = - \frac{x \cdot 3}{2} + 5 (\frac{3}{2})$

Step 2.3.1.5.2

Combine $5$ and $\frac{3}{2}$ .

$y - 5 = - \frac{x \cdot 3}{2} + \frac{5 \cdot 3}{2}$

Step 2.3.1.5.3

Multiply $5$ by $3$ .

$y - 5 = - \frac{x \cdot 3}{2} + \frac{15}{2}$

Step 2.3.1.6

Move $3$ to the left of $x$ .

$y - 5 = - \frac{3 x}{2} + \frac{15}{2}$

Step 2.3.2

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

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Step 2.3.2.1

Add $5$ to both sides of the equation.

$y = - \frac{3 x}{2} + \frac{15}{2} + 5$

Step 2.3.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} + 5 \cdot \frac{2}{2}$

Step 2.3.2.3

Combine $5$ and $\frac{2}{2}$ .

$y = - \frac{3 x}{2} + \frac{15}{2} + \frac{5 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{3 x}{2} + \frac{15 + 5 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $5$ by $2$ .

$y = - \frac{3 x}{2} + \frac{15 + 10}{2}$

Step 2.3.2.5.2

Add $15$ and $10$ .

$y = - \frac{3 x}{2} + \frac{25}{2}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{3}{2} x) + \frac{25}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{3}{2} x + \frac{25}{2}$

Step 3