Calculus Examples

$y = \arctan (\frac{1}{4} x)$ ; $(4, \frac{π}{4})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = \frac{π}{4}$ 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 $f (x) = \arctan (x)$ and $g (x) = \frac{1}{4} x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $\frac{1}{4} x$ .

$\frac{d}{d u} [\arctan (u)] \frac{d}{d x} [\frac{1}{4} x]$

Step 1.1.2

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

$\frac{1}{1 + u^{2}} \frac{d}{d x} [\frac{1}{4} x]$

Step 1.1.3

Replace all occurrences of $u$ with $\frac{1}{4} x$ .

$\frac{1}{1 + {(\frac{1}{4} x)}^{2}} \frac{d}{d x} [\frac{1}{4} x]$

Step 1.2

Differentiate.

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Step 1.2.1

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

$\frac{1}{1 + {(\frac{x}{4})}^{2}} \frac{d}{d x} [\frac{1}{4} x]$

Step 1.2.2

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

$\frac{1}{1 + {(\frac{x}{4})}^{2}} \frac{d}{d x} [\frac{x}{4}]$

Step 1.2.3

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$\frac{1}{1 + {(\frac{x}{4})}^{2}} (\frac{1}{4} \frac{d}{d x} [x])$

Step 1.2.4

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

$\frac{1}{4 (1 + {(\frac{x}{4})}^{2})} \frac{d}{d x} [x]$

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.3

Simplify.

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Step 1.3.1

Apply the product rule to $\frac{x}{4}$ .

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Combine terms.

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Step 1.3.3.1

Multiply $4$ by $1$ .

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

Step 1.3.3.2

Raise $4$ to the power of $2$ .

$\frac{1}{4 + 4 \frac{x^{2}}{16}}$

Step 1.3.3.3

Combine $4$ and $\frac{x^{2}}{16}$ .

$\frac{1}{4 + \frac{4 x^{2}}{16}}$

Step 1.3.3.4

Cancel the common factor of $4$ and $16$ .

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Step 1.3.3.4.1

Factor $4$ out of $4 x^{2}$ .

$\frac{1}{4 + \frac{4 (x^{2})}{16}}$

Step 1.3.3.4.2

Cancel the common factors.

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Step 1.3.3.4.2.1

Factor $4$ out of $16$ .

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

Step 1.3.3.4.2.2

Cancel the common factor.

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

Step 1.3.3.4.2.3

Rewrite the expression.

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

Step 1.3.4

Reorder terms.

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

Step 1.3.5

Simplify the denominator.

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Step 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

Combine the numerators over the common denominator.

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

Step 1.3.5.4

Multiply $4$ by $4$ .

$\frac{1}{\frac{x^{2} + 16}{4}}$

Step 1.3.6

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{4}{x^{2} + 16}$

Step 1.3.7

Multiply $\frac{4}{x^{2} + 16}$ by $1$ .

$\frac{4}{x^{2} + 16}$

Step 1.4

Evaluate the derivative at $x = 4$ .

$\frac{4}{{(4)}^{2} + 16}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Raise $4$ to the power of $2$ .

$\frac{4}{16 + 16}$

Step 1.5.1.2

Add $16$ and $16$ .

$\frac{4}{32}$

Step 1.5.2

Cancel the common factor of $4$ and $32$ .

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Step 1.5.2.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{32}$

Step 1.5.2.2

Cancel the common factors.

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Step 1.5.2.2.1

Factor $4$ out of $32$ .

$\frac{4 \cdot 1}{4 \cdot 8}$

Step 1.5.2.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 8}$

Step 1.5.2.2.3

Rewrite the expression.

$\frac{1}{8}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{8} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $\frac{1}{8}$ and $x$ .

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

Step 2.3.1.5

Cancel the common factor of $4$ .

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Step 2.3.1.5.1

Factor $4$ out of $8$ .

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

Step 2.3.1.5.2

Factor $4$ out of $- 4$ .

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

Step 2.3.1.5.3

Cancel the common factor.

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

Step 2.3.1.5.4

Rewrite the expression.

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

Step 2.3.1.6

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

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

Step 2.3.1.7

Move the negative in front of the fraction.

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

Step 2.3.2

Add $\frac{π}{4}$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

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

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

Step 2.3.3.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Multiply $2$ by $2$ .

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

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = \frac{x}{8} + \frac{- 2 + π}{4}$

Step 2.3.3.4

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

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

Step 2.3.3.5

Factor $- 1$ out of $π$ .

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

Step 2.3.3.6

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

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

Step 2.3.3.7

Move the negative in front of the fraction.

$y = \frac{x}{8} - \frac{2 - π}{4}$

Step 2.3.3.8

Reorder terms.

$y = \frac{1}{8} x - \frac{2 - π}{4}$

Step 3