Calculus Examples

$y = \ln (x^{2} - 3 x + 1)$ , $(3, 0)$

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 $g (x) = x^{2} - 3 x + 1$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 3 x + 1$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 1.1.2

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

$\frac{1}{u} \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{2} - 3 x + 1$ .

$\frac{1}{x^{2} - 3 x + 1} \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

$\frac{1}{x^{2} - 3 x + 1} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1])$

Step 1.2.2

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

$\frac{1}{x^{2} - 3 x + 1} (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1])$

Step 1.2.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$\frac{1}{x^{2} - 3 x + 1} (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 1.2.4

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

$\frac{1}{x^{2} - 3 x + 1} (2 x - 3 \cdot 1 + \frac{d}{d x} [1])$

Step 1.2.5

Multiply $- 3$ by $1$ .

$\frac{1}{x^{2} - 3 x + 1} (2 x - 3 + \frac{d}{d x} [1])$

Step 1.2.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{x^{2} - 3 x + 1} (2 x - 3 + 0)$

Step 1.2.7

Add $2 x - 3$ and $0$ .

$\frac{1}{x^{2} - 3 x + 1} (2 x - 3)$

Step 1.3

Simplify.

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Step 1.3.1

Reorder the factors of $\frac{1}{x^{2} - 3 x + 1} (2 x - 3)$ .

$(2 x - 3) \frac{1}{x^{2} - 3 x + 1}$

Step 1.3.2

Multiply $2 x - 3$ by $\frac{1}{x^{2} - 3 x + 1}$ .

$\frac{2 x - 3}{x^{2} - 3 x + 1}$

Step 1.4

Evaluate the derivative at $x = 3$ .

$\frac{2 (3) - 3}{{(3)}^{2} - 3 \cdot 3 + 1}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Multiply $2$ by $3$ .

$\frac{6 - 3}{3^{2} - 3 \cdot 3 + 1}$

Step 1.5.1.2

Subtract $3$ from $6$ .

$\frac{3}{3^{2} - 3 \cdot 3 + 1}$

Step 1.5.2

Simplify the denominator.

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Step 1.5.2.1

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

$\frac{3}{9 - 3 \cdot 3 + 1}$

Step 1.5.2.2

Multiply $- 3$ by $3$ .

$\frac{3}{9 - 9 + 1}$

Step 1.5.2.3

Subtract $9$ from $9$ .

$\frac{3}{0 + 1}$

Step 1.5.2.4

Add $0$ and $1$ .

$\frac{3}{1}$

Step 1.5.3

Divide $3$ by $1$ .

$3$

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 $3$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

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

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

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

Step 2.3.2.2

Multiply $3$ by $- 3$ .

$y = 3 x - 9$

Step 3