Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = 2$ 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 $g (x) = x^{2} - 2 x + 1$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Differentiate.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Multiply $- 2$ by $1$ .

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.3

Simplify.

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Step 1.3.1

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

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

Step 1.3.2

Factor using the perfect square rule.

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Step 1.3.2.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.3.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.3.2.3

Rewrite the polynomial.

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

Step 1.3.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 1.3.3

Multiply $2 x - 2$ by $\frac{1}{{(x - 1)}^{2}}$ .

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

Step 1.3.4

Factor $2$ out of $2 x - 2$ .

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Step 1.3.4.1

Factor $2$ out of $2 x$ .

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

Step 1.3.4.2

Factor $2$ out of $- 2$ .

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

Step 1.3.4.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

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

Step 1.3.5

Cancel the common factor of $x - 1$ and ${(x - 1)}^{2}$ .

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Step 1.3.5.1

Factor $x - 1$ out of $2 (x - 1)$ .

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

Step 1.3.5.2

Cancel the common factors.

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Step 1.3.5.2.1

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

$\frac{(x - 1) \cdot 2}{(x - 1) (x - 1)}$

Step 1.3.5.2.2

Cancel the common factor.

$\frac{(x - 1) \cdot 2}{(x - 1) (x - 1)}$

Step 1.3.5.2.3

Rewrite the expression.

$\frac{2}{x - 1}$

Step 1.4

Evaluate the derivative at $x = 2$ .

$\frac{2}{(2) - 1}$

Step 1.5

Simplify.

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Step 1.5.1

Subtract $1$ from $2$ .

$\frac{2}{1}$

Step 1.5.2

Divide $2$ by $1$ .

$2$

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

$y - (0) = 2 \cdot (x - (2))$

Step 2.2

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

$y + 0 = 2 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 2 \cdot (x - 2)$

Step 2.3.2

Simplify $2 \cdot (x - 2)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = 2 x + 2 \cdot - 2$

Step 2.3.2.2

Multiply $2$ by $- 2$ .

$y = 2 x - 4$

Step 3