Calculus Examples

$f (x) = \frac{10 x}{x^{2} - 6}$ ; $(4, 4)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 4$ to find the slope of the tangent line.

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Step 1.1

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10 x}{x^{2} - 6}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{x}{x^{2} - 6}]$ .

$10 \frac{d}{d x} [\frac{x}{x^{2} - 6}]$

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 $g (x) = x^{2} - 6$ .

$10 \frac{(x^{2} - 6) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 6]}{{(x^{2} - 6)}^{2}}$

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 $n = 1$ .

$10 \frac{(x^{2} - 6) \cdot 1 - x \frac{d}{d x} [x^{2} - 6]}{{(x^{2} - 6)}^{2}}$

Step 1.3.2

Multiply $x^{2} - 6$ by $1$ .

$10 \frac{x^{2} - 6 - x \frac{d}{d x} [x^{2} - 6]}{{(x^{2} - 6)}^{2}}$

Step 1.3.3

By the Sum Rule, the derivative of $x^{2} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]$ .

$10 \frac{x^{2} - 6 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6])}{{(x^{2} - 6)}^{2}}$

Step 1.3.4

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

$10 \frac{x^{2} - 6 - x (2 x + \frac{d}{d x} [- 6])}{{(x^{2} - 6)}^{2}}$

Step 1.3.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$10 \frac{x^{2} - 6 - x (2 x + 0)}{{(x^{2} - 6)}^{2}}$

Step 1.3.6

Simplify the expression.

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Step 1.3.6.1

Add $2 x$ and $0$ .

$10 \frac{x^{2} - 6 - x (2 x)}{{(x^{2} - 6)}^{2}}$

Step 1.3.6.2

Multiply $2$ by $- 1$ .

$10 \frac{x^{2} - 6 - 2 x \cdot x}{{(x^{2} - 6)}^{2}}$

Step 1.4

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

$10 \frac{x^{2} - 6 - 2 (x^{1} x)}{{(x^{2} - 6)}^{2}}$

Step 1.5

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

$10 \frac{x^{2} - 6 - 2 (x^{1} x^{1})}{{(x^{2} - 6)}^{2}}$

Step 1.6

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

$10 \frac{x^{2} - 6 - 2 x^{1 + 1}}{{(x^{2} - 6)}^{2}}$

Step 1.7

Add $1$ and $1$ .

$10 \frac{x^{2} - 6 - 2 x^{2}}{{(x^{2} - 6)}^{2}}$

Step 1.8

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

$10 \frac{- x^{2} - 6}{{(x^{2} - 6)}^{2}}$

Step 1.9

Combine $10$ and $\frac{- x^{2} - 6}{{(x^{2} - 6)}^{2}}$ .

$\frac{10 (- x^{2} - 6)}{{(x^{2} - 6)}^{2}}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

$\frac{10 (- x^{2}) + 10 \cdot - 6}{{(x^{2} - 6)}^{2}}$

Step 1.10.2

Simplify each term.

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Step 1.10.2.1

Multiply $- 1$ by $10$ .

$\frac{- 10 x^{2} + 10 \cdot - 6}{{(x^{2} - 6)}^{2}}$

Step 1.10.2.2

Multiply $10$ by $- 6$ .

$\frac{- 10 x^{2} - 60}{{(x^{2} - 6)}^{2}}$

Step 1.10.3

Factor $10$ out of $- 10 x^{2} - 60$ .

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Step 1.10.3.1

Factor $10$ out of $- 10 x^{2}$ .

$\frac{10 (- x^{2}) - 60}{{(x^{2} - 6)}^{2}}$

Step 1.10.3.2

Factor $10$ out of $- 60$ .

$\frac{10 (- x^{2}) + 10 (- 6)}{{(x^{2} - 6)}^{2}}$

Step 1.10.3.3

Factor $10$ out of $10 (- x^{2}) + 10 (- 6)$ .

$\frac{10 (- x^{2} - 6)}{{(x^{2} - 6)}^{2}}$

Step 1.10.4

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

$\frac{10 (- (x^{2}) - 6)}{{(x^{2} - 6)}^{2}}$

Step 1.10.5

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{10 (- (x^{2}) - 1 (6))}{{(x^{2} - 6)}^{2}}$

Step 1.10.6

Factor $- 1$ out of $- (x^{2}) - 1 (6)$ .

$\frac{10 (- (x^{2} + 6))}{{(x^{2} - 6)}^{2}}$

Step 1.10.7

Rewrite $- (x^{2} + 6)$ as $- 1 (x^{2} + 6)$ .

$\frac{10 (- 1 (x^{2} + 6))}{{(x^{2} - 6)}^{2}}$

Step 1.10.8

Move the negative in front of the fraction.

$- \frac{10 (x^{2} + 6)}{{(x^{2} - 6)}^{2}}$

Step 1.11

Evaluate the derivative at $x = 4$ .

$- \frac{10 ({(4)}^{2} + 6)}{{({(4)}^{2} - 6)}^{2}}$

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 $4$ to the power of $2$ .

$- \frac{10 (16 + 6)}{{(4^{2} - 6)}^{2}}$

Step 1.12.1.2

Add $16$ and $6$ .

$- \frac{10 \cdot 22}{{(4^{2} - 6)}^{2}}$

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

Raise $4$ to the power of $2$ .

$- \frac{10 \cdot 22}{{(16 - 6)}^{2}}$

Step 1.12.2.2

Subtract $6$ from $16$ .

$- \frac{10 \cdot 22}{10^{2}}$

Step 1.12.2.3

Raise $10$ to the power of $2$ .

$- \frac{10 \cdot 22}{100}$

Step 1.12.3

Reduce the expression by cancelling the common factors.

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Step 1.12.3.1

Multiply $10$ by $22$ .

$- \frac{220}{100}$

Step 1.12.3.2

Cancel the common factor of $220$ and $100$ .

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Step 1.12.3.2.1

Factor $20$ out of $220$ .

$- \frac{20 (11)}{100}$

Step 1.12.3.2.2

Cancel the common factors.

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Step 1.12.3.2.2.1

Factor $20$ out of $100$ .

$- \frac{20 \cdot 11}{20 \cdot 5}$

Step 1.12.3.2.2.2

Cancel the common factor.

$- \frac{20 \cdot 11}{20 \cdot 5}$

Step 1.12.3.2.2.3

Rewrite the expression.

$- \frac{11}{5}$

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 $- \frac{11}{5}$ and a given point $(4, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = - \frac{11}{5} \cdot (x - (4))$

Step 2.2

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

$y - 4 = - \frac{11}{5} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{11}{5} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 - \frac{11}{5} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = - \frac{11}{5} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = - \frac{11}{5} x - \frac{11}{5} \cdot - 4$

Step 2.3.1.4

Combine $x$ and $\frac{11}{5}$ .

$y - 4 = - \frac{x \cdot 11}{5} - \frac{11}{5} \cdot - 4$

Step 2.3.1.5

Multiply $- \frac{11}{5} \cdot - 4$ .

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Step 2.3.1.5.1

Multiply $- 4$ by $- 1$ .

$y - 4 = - \frac{x \cdot 11}{5} + 4 (\frac{11}{5})$

Step 2.3.1.5.2

Combine $4$ and $\frac{11}{5}$ .

$y - 4 = - \frac{x \cdot 11}{5} + \frac{4 \cdot 11}{5}$

Step 2.3.1.5.3

Multiply $4$ by $11$ .

$y - 4 = - \frac{x \cdot 11}{5} + \frac{44}{5}$

Step 2.3.1.6

Move $11$ to the left of $x$ .

$y - 4 = - \frac{11 x}{5} + \frac{44}{5}$

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 $4$ to both sides of the equation.

$y = - \frac{11 x}{5} + \frac{44}{5} + 4$

Step 2.3.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = - \frac{11 x}{5} + \frac{44}{5} + 4 \cdot \frac{5}{5}$

Step 2.3.2.3

Combine $4$ and $\frac{5}{5}$ .

$y = - \frac{11 x}{5} + \frac{44}{5} + \frac{4 \cdot 5}{5}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{11 x}{5} + \frac{44 + 4 \cdot 5}{5}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $4$ by $5$ .

$y = - \frac{11 x}{5} + \frac{44 + 20}{5}$

Step 2.3.2.5.2

Add $44$ and $20$ .

$y = - \frac{11 x}{5} + \frac{64}{5}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{11}{5} x) + \frac{64}{5}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{11}{5} x + \frac{64}{5}$

Step 3