Calculus Examples

$y = \frac{18}{x^{2} + 2}$ , $(1, 6)$

Step 1

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

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $18$ is constant with respect to $x$ , the derivative of $\frac{18}{x^{2} + 2}$ with respect to $x$ is $18 \frac{d}{d x} [\frac{1}{x^{2} + 2}]$ .

$18 \frac{d}{d x} [\frac{1}{x^{2} + 2}]$

Step 1.1.2

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

$18 \frac{d}{d x} [{(x^{2} + 2)}^{- 1}]$

Step 1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2} + 2$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 2$ .

$18 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 2])$

Step 1.2.2

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

$18 (- u^{- 2} \frac{d}{d x} [x^{2} + 2])$

Step 1.2.3

Replace all occurrences of $u$ with $x^{2} + 2$ .

$18 (- {(x^{2} + 2)}^{- 2} \frac{d}{d x} [x^{2} + 2])$

Step 1.3

Differentiate.

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Step 1.3.1

Multiply $- 1$ by $18$ .

$- 18 ({(x^{2} + 2)}^{- 2} \frac{d}{d x} [x^{2} + 2])$

Step 1.3.2

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

$- 18 {(x^{2} + 2)}^{- 2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2])$

Step 1.3.3

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

$- 18 {(x^{2} + 2)}^{- 2} (2 x + \frac{d}{d x} [2])$

Step 1.3.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$- 18 {(x^{2} + 2)}^{- 2} (2 x + 0)$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $2 x$ and $0$ .

$- 18 {(x^{2} + 2)}^{- 2} (2 x)$

Step 1.3.5.2

Multiply $2$ by $- 18$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4.2

Combine terms.

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Step 1.4.2.1

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

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

Step 1.4.2.2

Move the negative in front of the fraction.

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

Step 1.4.2.3

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

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

Step 1.4.2.4

Move $36$ to the left of $x$ .

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

Step 1.5

Evaluate the derivative at $x = 1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Multiply $36$ by $1$ .

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

Step 1.6.2

Simplify the denominator.

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Step 1.6.2.1

One to any power is one.

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

Step 1.6.2.2

Add $1$ and $2$ .

$- \frac{36}{3^{2}}$

Step 1.6.2.3

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

$- \frac{36}{9}$

Step 1.6.3

Simplify the expression.

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Step 1.6.3.1

Divide $36$ by $9$ .

$- 1 \cdot 4$

Step 1.6.3.2

Multiply $- 1$ by $4$ .

$- 4$

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 $- 4$ and a given point $(1, 6)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (6) = - 4 \cdot (x - (1))$

Step 2.2

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

$y - 6 = - 4 \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 4 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 6 = 0 + 0 - 4 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 6 = - 4 \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 6 = - 4 x - 4 \cdot - 1$

Step 2.3.1.4

Multiply $- 4$ by $- 1$ .

$y - 6 = - 4 x + 4$

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 $6$ to both sides of the equation.

$y = - 4 x + 4 + 6$

Step 2.3.2.2

Add $4$ and $6$ .

$y = - 4 x + 10$

Step 3