Calculus Examples

$y = \ln (x^{2} - 8)$ , $(3, 0)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{2} - 8$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 8$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} - 8]$

Step 1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x^{2} - 8]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} - 8$ .

$\frac{1}{x^{2} - 8} \frac{d}{d x} [x^{2} - 8]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8]$ .

$\frac{1}{x^{2} - 8} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8])$

Step 1.2.2

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

$\frac{1}{x^{2} - 8} (2 x + \frac{d}{d x} [- 8])$

Step 1.2.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{1}{x^{2} - 8} (2 x + 0)$

Step 1.2.4

Combine fractions.

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Step 1.2.4.1

Add $2 x$ and $0$ .

$\frac{1}{x^{2} - 8} (2 x)$

Step 1.2.4.2

Combine $2$ and $\frac{1}{x^{2} - 8}$ .

$\frac{2}{x^{2} - 8} x$

Step 1.2.4.3

Combine $\frac{2}{x^{2} - 8}$ and $x$ .

$\frac{2 x}{x^{2} - 8}$

Step 1.3

Evaluate the derivative at $x = 3$ .

$\frac{2 (3)}{{(3)}^{2} - 8}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $2$ by $3$ .

$\frac{6}{3^{2} - 8}$

Step 1.4.2

Simplify the denominator.

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Step 1.4.2.1

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

$\frac{6}{9 - 8}$

Step 1.4.2.2

Subtract $8$ from $9$ .

$\frac{6}{1}$

Step 1.4.3

Divide $6$ by $1$ .

$6$

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 $6$ and a given point $(3, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 6 \cdot (x - (3))$

Step 2.2

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

$y + 0 = 6 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 6 \cdot (x - 3)$

Step 2.3.2

Simplify $6 \cdot (x - 3)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = 6 x + 6 \cdot - 3$

Step 2.3.2.2

Multiply $6$ by $- 3$ .

$y = 6 x - 18$

Step 3