Calculus Examples

$\frac{36 x}{x^{2} + 36}$ , $(- 3, - \frac{12}{5})$

Step 1

Write $\frac{36 x}{x^{2} + 36}$ as an equation.

$y = \frac{36 x}{x^{2} + 36}$

Step 2

Find the first derivative and evaluate at $x = - 3$ and $y = - \frac{12}{5}$ to find the slope of the tangent line.

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Step 2.1

Since $36$ is constant with respect to $x$ , the derivative of $\frac{36 x}{x^{2} + 36}$ with respect to $x$ is $36 \frac{d}{d x} [\frac{x}{x^{2} + 36}]$ .

$36 \frac{d}{d x} [\frac{x}{x^{2} + 36}]$

Step 2.2

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 $g (x) = x^{2} + 36$ .

$36 \frac{(x^{2} + 36) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.3

Differentiate.

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Step 2.3.1

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

$36 \frac{(x^{2} + 36) \cdot 1 - x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.3.2

Multiply $x^{2} + 36$ by $1$ .

$36 \frac{x^{2} + 36 - x \frac{d}{d x} [x^{2} + 36]}{{(x^{2} + 36)}^{2}}$

Step 2.3.3

By the Sum Rule, the derivative of $x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$ .

$36 \frac{x^{2} + 36 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.3.4

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

$36 \frac{x^{2} + 36 - x (2 x + \frac{d}{d x} [36])}{{(x^{2} + 36)}^{2}}$

Step 2.3.5

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$36 \frac{x^{2} + 36 - x (2 x + 0)}{{(x^{2} + 36)}^{2}}$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Add $2 x$ and $0$ .

$36 \frac{x^{2} + 36 - x (2 x)}{{(x^{2} + 36)}^{2}}$

Step 2.3.6.2

Multiply $2$ by $- 1$ .

$36 \frac{x^{2} + 36 - 2 x \cdot x}{{(x^{2} + 36)}^{2}}$

Step 2.4

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

$36 \frac{x^{2} + 36 - 2 (x^{1} x)}{{(x^{2} + 36)}^{2}}$

Step 2.5

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

$36 \frac{x^{2} + 36 - 2 (x^{1} x^{1})}{{(x^{2} + 36)}^{2}}$

Step 2.6

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

$36 \frac{x^{2} + 36 - 2 x^{1 + 1}}{{(x^{2} + 36)}^{2}}$

Step 2.7

Add $1$ and $1$ .

$36 \frac{x^{2} + 36 - 2 x^{2}}{{(x^{2} + 36)}^{2}}$

Step 2.8

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

$36 \frac{- x^{2} + 36}{{(x^{2} + 36)}^{2}}$

Step 2.9

Combine $36$ and $\frac{- x^{2} + 36}{{(x^{2} + 36)}^{2}}$ .

$\frac{36 (- x^{2} + 36)}{{(x^{2} + 36)}^{2}}$

Step 2.10

Simplify.

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Step 2.10.1

Apply the distributive property.

$\frac{36 (- x^{2}) + 36 \cdot 36}{{(x^{2} + 36)}^{2}}$

Step 2.10.2

Simplify each term.

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Step 2.10.2.1

Multiply $- 1$ by $36$ .

$\frac{- 36 x^{2} + 36 \cdot 36}{{(x^{2} + 36)}^{2}}$

Step 2.10.2.2

Multiply $36$ by $36$ .

$\frac{- 36 x^{2} + 1296}{{(x^{2} + 36)}^{2}}$

Step 2.11

Evaluate the derivative at $x = - 3$ .

$\frac{- 36 {(- 3)}^{2} + 1296}{{({(- 3)}^{2} + 36)}^{2}}$

Step 2.12

Simplify.

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Step 2.12.1

Simplify the numerator.

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Step 2.12.1.1

Raise $- 3$ to the power of $2$ .

$\frac{- 36 \cdot 9 + 1296}{{({(- 3)}^{2} + 36)}^{2}}$

Step 2.12.1.2

Multiply $- 36$ by $9$ .

$\frac{- 324 + 1296}{{({(- 3)}^{2} + 36)}^{2}}$

Step 2.12.1.3

Add $- 324$ and $1296$ .

$\frac{972}{{({(- 3)}^{2} + 36)}^{2}}$

Step 2.12.2

Simplify the denominator.

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Step 2.12.2.1

Raise $- 3$ to the power of $2$ .

$\frac{972}{{(9 + 36)}^{2}}$

Step 2.12.2.2

Add $9$ and $36$ .

$\frac{972}{45^{2}}$

Step 2.12.2.3

Raise $45$ to the power of $2$ .

$\frac{972}{2025}$

Step 2.12.3

Cancel the common factor of $972$ and $2025$ .

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Step 2.12.3.1

Factor $81$ out of $972$ .

$\frac{81 (12)}{2025}$

Step 2.12.3.2

Cancel the common factors.

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Step 2.12.3.2.1

Factor $81$ out of $2025$ .

$\frac{81 \cdot 12}{81 \cdot 25}$

Step 2.12.3.2.2

Cancel the common factor.

$\frac{81 \cdot 12}{81 \cdot 25}$

Step 2.12.3.2.3

Rewrite the expression.

$\frac{12}{25}$

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 $\frac{12}{25}$ and a given point $(- 3, - \frac{12}{5})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{12}{5}) = \frac{12}{25} \cdot (x - (- 3))$

Step 3.2

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

$y + \frac{12}{5} = \frac{12}{25} \cdot (x + 3)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{12}{25} \cdot (x + 3)$ .

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Step 3.3.1.1

Rewrite.

$y + \frac{12}{5} = 0 + 0 + \frac{12}{25} \cdot (x + 3)$

Step 3.3.1.2

Simplify by adding zeros.

$y + \frac{12}{5} = \frac{12}{25} \cdot (x + 3)$

Step 3.3.1.3

Apply the distributive property.

$y + \frac{12}{5} = \frac{12}{25} x + \frac{12}{25} \cdot 3$

Step 3.3.1.4

Combine $\frac{12}{25}$ and $x$ .

$y + \frac{12}{5} = \frac{12 x}{25} + \frac{12}{25} \cdot 3$

Step 3.3.1.5

Multiply $\frac{12}{25} \cdot 3$ .

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Step 3.3.1.5.1

Combine $\frac{12}{25}$ and $3$ .

$y + \frac{12}{5} = \frac{12 x}{25} + \frac{12 \cdot 3}{25}$

Step 3.3.1.5.2

Multiply $12$ by $3$ .

$y + \frac{12}{5} = \frac{12 x}{25} + \frac{36}{25}$

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

Subtract $\frac{12}{5}$ from both sides of the equation.

$y = \frac{12 x}{25} + \frac{36}{25} - \frac{12}{5}$

Step 3.3.2.2

To write $- \frac{12}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$y = \frac{12 x}{25} + \frac{36}{25} - \frac{12}{5} \cdot \frac{5}{5}$

Step 3.3.2.3

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.3.1

Multiply $\frac{12}{5}$ by $\frac{5}{5}$ .

$y = \frac{12 x}{25} + \frac{36}{25} - \frac{12 \cdot 5}{5 \cdot 5}$

Step 3.3.2.3.2

Multiply $5$ by $5$ .

$y = \frac{12 x}{25} + \frac{36}{25} - \frac{12 \cdot 5}{25}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$y = \frac{12 x}{25} + \frac{36 - 12 \cdot 5}{25}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply $- 12$ by $5$ .

$y = \frac{12 x}{25} + \frac{36 - 60}{25}$

Step 3.3.2.5.2

Subtract $60$ from $36$ .

$y = \frac{12 x}{25} + \frac{- 24}{25}$

Step 3.3.2.6

Move the negative in front of the fraction.

$y = \frac{12 x}{25} - \frac{24}{25}$

Step 3.3.3

Reorder terms.

$y = \frac{12}{25} x - \frac{24}{25}$

Step 4