Calculus Examples

$y = \frac{1}{2} \arctan (x)$ ; $(1, \frac{π}{8})$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = \frac{π}{8}$ to find the slope of the tangent line.

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Step 1.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \cdot \arctan (x)$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\arctan (x)]$ .

$\frac{1}{2} \frac{d}{d x} [\arctan (x)]$

Step 1.2

The derivative of $\arctan (x)$ with respect to $x$ is $\frac{1}{1 + x^{2}}$ .

$\frac{1}{2} \cdot \frac{1}{1 + x^{2}}$

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$\frac{1}{2 \cdot 1 + 2 x^{2}}$

Step 1.4.2

Multiply $2$ by $1$ .

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

Step 1.4.3

Reorder terms.

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

Step 1.4.4

Factor $2$ out of $2 x^{2} + 2$ .

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Step 1.4.4.1

Factor $2$ out of $2 x^{2}$ .

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

Step 1.4.4.2

Factor $2$ out of $2$ .

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

Step 1.4.4.3

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

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

Step 1.5

Evaluate the derivative at $x = 1$ .

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

Step 1.6

Simplify.

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Step 1.6.1

Simplify the denominator.

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Step 1.6.1.1

One to any power is one.

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

Step 1.6.1.2

Add $1$ and $1$ .

$\frac{1}{2 \cdot 2}$

Step 1.6.2

Multiply $2$ by $2$ .

$\frac{1}{4}$

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 $\frac{1}{4}$ and a given point $(1, \frac{π}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{8}) = \frac{1}{4} \cdot (x - (1))$

Step 2.2

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

$y - \frac{π}{8} = \frac{1}{4} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{4} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{π}{8} = 0 + 0 + \frac{1}{4} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{π}{8} = \frac{1}{4} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{π}{8} = \frac{1}{4} x + \frac{1}{4} \cdot - 1$

Step 2.3.1.4

Combine $\frac{1}{4}$ and $x$ .

$y - \frac{π}{8} = \frac{x}{4} + \frac{1}{4} \cdot - 1$

Step 2.3.1.5

Combine $\frac{1}{4}$ and $- 1$ .

$y - \frac{π}{8} = \frac{x}{4} + \frac{- 1}{4}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - \frac{π}{8} = \frac{x}{4} - \frac{1}{4}$

Step 2.3.2

Add $\frac{π}{8}$ to both sides of the equation.

$y = \frac{x}{4} - \frac{1}{4} + \frac{π}{8}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{4} - \frac{1}{4} \cdot \frac{2}{2} + \frac{π}{8}$

Step 2.3.3.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.2.1

Multiply $\frac{1}{4}$ by $\frac{2}{2}$ .

$y = \frac{x}{4} - \frac{2}{4 \cdot 2} + \frac{π}{8}$

Step 2.3.3.2.2

Multiply $4$ by $2$ .

$y = \frac{x}{4} - \frac{2}{8} + \frac{π}{8}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = \frac{x}{4} + \frac{- 2 + π}{8}$

Step 2.3.3.4

Rewrite $- 2$ as $- 1 (2)$ .

$y = \frac{x}{4} + \frac{- 1 (2) + π}{8}$

Step 2.3.3.5

Factor $- 1$ out of $π$ .

$y = \frac{x}{4} + \frac{- 1 (2) - 1 (- π)}{8}$

Step 2.3.3.6

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

$y = \frac{x}{4} + \frac{- 1 (2 - π)}{8}$

Step 2.3.3.7

Move the negative in front of the fraction.

$y = \frac{x}{4} - \frac{2 - π}{8}$

Step 2.3.3.8

Reorder terms.

$y = \frac{1}{4} x - \frac{2 - π}{8}$

Step 3