Calculus Examples

$y = \frac{1}{3} \arctan (\frac{1}{2} x)$ , $(2, \frac{π}{12})$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = \frac{π}{12}$ to find the slope of the tangent line.

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Step 1.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} \cdot \arctan (\frac{1}{2} x)$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\arctan (\frac{1}{2} x)]$ .

$\frac{1}{3} \frac{d}{d x} [\arctan (\frac{1}{2} x)]$

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 $f (x) = \arctan (x)$ and $g (x) = \frac{1}{2} x$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\frac{1}{2} x$ .

$\frac{1}{3} (\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{1}{2} x])$

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $\frac{1}{2} x$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 1.3.2

Combine fractions.

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Step 1.3.2.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Move $3$ to the left of $1 + {(\frac{x}{2})}^{2}$ .

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

Step 1.3.3

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

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

Step 1.3.4

Combine fractions.

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Step 1.3.4.1

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

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

Step 1.3.4.2

Multiply $3$ by $2$ .

$\frac{1}{6 (1 + {(\frac{x}{2})}^{2})} \frac{d}{d x} [x]$

Step 1.3.5

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

$\frac{1}{6 (1 + {(\frac{x}{2})}^{2})} \cdot 1$

Step 1.3.6

Multiply $\frac{1}{6 (1 + {(\frac{x}{2})}^{2})}$ by $1$ .

$\frac{1}{6 (1 + {(\frac{x}{2})}^{2})}$

Step 1.4

Simplify.

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Step 1.4.1

Apply the product rule to $\frac{x}{2}$ .

$\frac{1}{6 (1 + \frac{x^{2}}{2^{2}})}$

Step 1.4.2

Apply the distributive property.

$\frac{1}{6 \cdot 1 + 6 \frac{x^{2}}{2^{2}}}$

Step 1.4.3

Combine terms.

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Step 1.4.3.1

Multiply $6$ by $1$ .

$\frac{1}{6 + 6 \frac{x^{2}}{2^{2}}}$

Step 1.4.3.2

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

$\frac{1}{6 + 6 \frac{x^{2}}{4}}$

Step 1.4.3.3

Combine $6$ and $\frac{x^{2}}{4}$ .

$\frac{1}{6 + \frac{6 x^{2}}{4}}$

Step 1.4.3.4

Cancel the common factor of $6$ and $4$ .

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Step 1.4.3.4.1

Factor $2$ out of $6 x^{2}$ .

$\frac{1}{6 + \frac{2 (3 x^{2})}{4}}$

Step 1.4.3.4.2

Cancel the common factors.

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Step 1.4.3.4.2.1

Factor $2$ out of $4$ .

$\frac{1}{6 + \frac{2 (3 x^{2})}{2 (2)}}$

Step 1.4.3.4.2.2

Cancel the common factor.

$\frac{1}{6 + \frac{2 (3 x^{2})}{2 \cdot 2}}$

Step 1.4.3.4.2.3

Rewrite the expression.

$\frac{1}{6 + \frac{3 x^{2}}{2}}$

Step 1.4.4

Reorder terms.

$\frac{1}{\frac{3 x^{2}}{2} + 6}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Factor $3$ out of $\frac{3 x^{2}}{2} + 6$ .

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Step 1.4.5.1.1

Factor $3$ out of $\frac{3 x^{2}}{2}$ .

$\frac{1}{3 \frac{x^{2}}{2} + 6}$

Step 1.4.5.1.2

Factor $3$ out of $6$ .

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

Step 1.4.5.1.3

Factor $3$ out of $3 \frac{x^{2}}{2} + 3 \cdot 2$ .

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

Step 1.4.5.2

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

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

Step 1.4.5.3

Combine $2$ and $\frac{2}{2}$ .

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

Step 1.4.5.4

Combine the numerators over the common denominator.

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

Step 1.4.5.5

Multiply $2$ by $2$ .

$\frac{1}{3 \frac{x^{2} + 4}{2}}$

Step 1.4.6

Combine $3$ and $\frac{x^{2} + 4}{2}$ .

$\frac{1}{\frac{3 (x^{2} + 4)}{2}}$

Step 1.4.7

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2}{3 (x^{2} + 4)}$

Step 1.4.8

Multiply $\frac{2}{3 (x^{2} + 4)}$ by $1$ .

$\frac{2}{3 (x^{2} + 4)}$

Step 1.5

Evaluate the derivative at $x = 2$ .

$\frac{2}{3 ({(2)}^{2} + 4)}$

Step 1.6

Simplify.

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Step 1.6.1

Simplify the denominator.

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Step 1.6.1.1

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

$\frac{2}{3 (4 + 4)}$

Step 1.6.1.2

Add $4$ and $4$ .

$\frac{2}{3 \cdot 8}$

Step 1.6.2

Reduce the expression by cancelling the common factors.

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Step 1.6.2.1

Multiply $3$ by $8$ .

$\frac{2}{24}$

Step 1.6.2.2

Cancel the common factor of $2$ and $24$ .

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Step 1.6.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{24}$

Step 1.6.2.2.2

Cancel the common factors.

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Step 1.6.2.2.2.1

Factor $2$ out of $24$ .

$\frac{2 \cdot 1}{2 \cdot 12}$

Step 1.6.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 12}$

Step 1.6.2.2.2.3

Rewrite the expression.

$\frac{1}{12}$

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 $\frac{1}{12}$ and a given point $(2, \frac{π}{12})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{12}) = \frac{1}{12} \cdot (x - (2))$

Step 2.2

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

$y - \frac{π}{12} = \frac{1}{12} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{12} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{π}{12} = 0 + 0 + \frac{1}{12} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{π}{12} = \frac{1}{12} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{π}{12} = \frac{1}{12} x + \frac{1}{12} \cdot - 2$

Step 2.3.1.4

Combine $\frac{1}{12}$ and $x$ .

$y - \frac{π}{12} = \frac{x}{12} + \frac{1}{12} \cdot - 2$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $12$ .

$y - \frac{π}{12} = \frac{x}{12} + \frac{1}{2 (6)} \cdot - 2$

Step 2.3.1.5.2

Factor $2$ out of $- 2$ .

$y - \frac{π}{12} = \frac{x}{12} + \frac{1}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - \frac{π}{12} = \frac{x}{12} + \frac{1}{2 \cdot 6} \cdot (2 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - \frac{π}{12} = \frac{x}{12} + \frac{1}{6} \cdot - 1$

Step 2.3.1.6

Combine $\frac{1}{6}$ and $- 1$ .

$y - \frac{π}{12} = \frac{x}{12} + \frac{- 1}{6}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - \frac{π}{12} = \frac{x}{12} - \frac{1}{6}$

Step 2.3.2

Add $\frac{π}{12}$ to both sides of the equation.

$y = \frac{x}{12} - \frac{1}{6} + \frac{π}{12}$

Step 2.3.3

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

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Step 2.3.3.1

To write $- \frac{1}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{12} - \frac{1}{6} \cdot \frac{2}{2} + \frac{π}{12}$

Step 2.3.3.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.2.1

Multiply $\frac{1}{6}$ by $\frac{2}{2}$ .

$y = \frac{x}{12} - \frac{2}{6 \cdot 2} + \frac{π}{12}$

Step 2.3.3.2.2

Multiply $6$ by $2$ .

$y = \frac{x}{12} - \frac{2}{12} + \frac{π}{12}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = \frac{x}{12} + \frac{- 2 + π}{12}$

Step 2.3.3.4

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

$y = \frac{x}{12} + \frac{- 1 (2) + π}{12}$

Step 2.3.3.5

Factor $- 1$ out of $π$ .

$y = \frac{x}{12} + \frac{- 1 (2) - 1 (- π)}{12}$

Step 2.3.3.6

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

$y = \frac{x}{12} + \frac{- 1 (2 - π)}{12}$

Step 2.3.3.7

Move the negative in front of the fraction.

$y = \frac{x}{12} - \frac{2 - π}{12}$

Step 2.3.3.8

Reorder terms.

$y = \frac{1}{12} x - \frac{2 - π}{12}$

Step 3