Calculus Examples

$y = arcsec (6 x)$ , $(\frac{\sqrt{2}}{6}, \frac{π}{4})$

Step 1

Find the first derivative and evaluate at $x = \frac{\sqrt{2}}{6}$ 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 $f (x) = arcsec (x)$ and $g (x) = 6 x$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $6 x$ .

$\frac{d}{d u} [arcsec (u)] \frac{d}{d x} [6 x]$

Step 1.1.2

The derivative of $arcsec (u)$ with respect to $u$ is $\frac{1}{u \sqrt{u^{2} - 1}}$ .

$\frac{1}{u \sqrt{u^{2} - 1}} \frac{d}{d x} [6 x]$

Step 1.1.3

Replace all occurrences of $u$ with $6 x$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

Factor $6$ out of $6 x$ .

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

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Apply the product rule to $6 (x)$ .

$\frac{1}{6 x \sqrt{6^{2} x^{2} - 1}} \frac{d}{d x} [6 x]$

Step 1.2.2.2

Raise $6$ to the power of $2$ .

$\frac{1}{6 x \sqrt{36 x^{2} - 1}} \frac{d}{d x} [6 x]$

Step 1.2.3

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{1}{6 x \sqrt{36 x^{2} - 1}} (6 \frac{d}{d x} [x])$

Step 1.2.4

Simplify terms.

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Step 1.2.4.1

Combine $6$ and $\frac{1}{6 x \sqrt{36 x^{2} - 1}}$ .

$\frac{6}{6 x \sqrt{36 x^{2} - 1}} \frac{d}{d x} [x]$

Step 1.2.4.2

Cancel the common factor of $6$ .

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Step 1.2.4.2.1

Cancel the common factor.

$\frac{6}{6 x \sqrt{36 x^{2} - 1}} \frac{d}{d x} [x]$

Step 1.2.4.2.2

Rewrite the expression.

$\frac{1}{x \sqrt{36 x^{2} - 1}} \frac{d}{d x} [x]$

Step 1.2.5

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

$\frac{1}{x \sqrt{36 x^{2} - 1}} \cdot 1$

Step 1.2.6

Multiply $\frac{1}{x \sqrt{36 x^{2} - 1}}$ by $1$ .

$\frac{1}{x \sqrt{36 x^{2} - 1}}$

Step 1.3

Evaluate the derivative at $x = \frac{\sqrt{2}}{6}$ .

$\frac{1}{(\frac{\sqrt{2}}{6}) \sqrt{36 {(\frac{\sqrt{2}}{6})}^{2} - 1}}$

Step 1.4

Simplify.

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Step 1.4.1

Combine $\frac{\sqrt{2}}{6}$ and $\sqrt{36 {(\frac{\sqrt{2}}{6})}^{2} - 1}$ .

$\frac{1}{\frac{\sqrt{2} \sqrt{36 {(\frac{\sqrt{2}}{6})}^{2} - 1}}{6}}$

Step 1.4.2

Simplify the numerator.

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Step 1.4.2.1

Combine using the product rule for radicals.

$\frac{1}{\frac{\sqrt{2 (36 {(\frac{\sqrt{2}}{6})}^{2} - 1)}}{6}}$

Step 1.4.2.2

Apply the product rule to $\frac{\sqrt{2}}{6}$ .

$\frac{1}{\frac{\sqrt{2 (36 \frac{{\sqrt{2}}^{2}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{1}{\frac{\sqrt{2 (36 \frac{{(2^{\frac{1}{2}})}^{2}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\frac{\sqrt{2 (36 \frac{2^{\frac{1}{2} \cdot 2}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3.3

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

$\frac{1}{\frac{\sqrt{2 (36 \frac{2^{\frac{2}{2}}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3.4

Cancel the common factor of $2$ .

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Step 1.4.2.3.4.1

Cancel the common factor.

$\frac{1}{\frac{\sqrt{2 (36 \frac{2^{\frac{2}{2}}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3.4.2

Rewrite the expression.

$\frac{1}{\frac{\sqrt{2 (36 \frac{2^{1}}{6^{2}} - 1)}}{6}}$

Step 1.4.2.3.5

Evaluate the exponent.

$\frac{1}{\frac{\sqrt{2 (36 \frac{2}{6^{2}} - 1)}}{6}}$

Step 1.4.2.4

Raise $6$ to the power of $2$ .

$\frac{1}{\frac{\sqrt{2 (36 (\frac{2}{36}) - 1)}}{6}}$

Step 1.4.2.5

Cancel the common factor of $36$ .

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Step 1.4.2.5.1

Cancel the common factor.

$\frac{1}{\frac{\sqrt{2 (36 (\frac{2}{36}) - 1)}}{6}}$

Step 1.4.2.5.2

Rewrite the expression.

$\frac{1}{\frac{\sqrt{2 (2 - 1)}}{6}}$

Step 1.4.2.6

Subtract $1$ from $2$ .

$\frac{1}{\frac{\sqrt{2 \cdot 1}}{6}}$

Step 1.4.3

Multiply $2$ by $1$ .

$\frac{1}{\frac{\sqrt{2}}{6}}$

Step 1.4.4

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{6}{\sqrt{2}}$

Step 1.4.5

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

$\frac{6}{\sqrt{2}}$

Step 1.4.6

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

$\frac{6}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4.7

Combine and simplify the denominator.

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Step 1.4.7.1

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

$\frac{6 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{6 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{6 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.4.7.5

Add $1$ and $1$ .

$\frac{6 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{6 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{6 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.4.7.6.3

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

$\frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.7.6.4

Cancel the common factor of $2$ .

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Step 1.4.7.6.4.1

Cancel the common factor.

$\frac{6 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.7.6.4.2

Rewrite the expression.

$\frac{6 \sqrt{2}}{2^{1}}$

Step 1.4.7.6.5

Evaluate the exponent.

$\frac{6 \sqrt{2}}{2}$

Step 1.4.8

Cancel the common factor of $6$ and $2$ .

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Step 1.4.8.1

Factor $2$ out of $6 \sqrt{2}$ .

$\frac{2 (3 \sqrt{2})}{2}$

Step 1.4.8.2

Cancel the common factors.

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Step 1.4.8.2.1

Factor $2$ out of $2$ .

$\frac{2 (3 \sqrt{2})}{2 (1)}$

Step 1.4.8.2.2

Cancel the common factor.

$\frac{2 (3 \sqrt{2})}{2 \cdot 1}$

Step 1.4.8.2.3

Rewrite the expression.

$\frac{3 \sqrt{2}}{1}$

Step 1.4.8.2.4

Divide $3 \sqrt{2}$ by $1$ .

$3 \sqrt{2}$

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 $3 \sqrt{2}$ and a given point $(\frac{\sqrt{2}}{6}, \frac{π}{4})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{4}) = 3 \sqrt{2} \cdot (x - (\frac{\sqrt{2}}{6}))$

Step 2.2

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

$y - \frac{π}{4} = 3 \sqrt{2} \cdot (x - \frac{\sqrt{2}}{6})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $3 \sqrt{2} \cdot (x - \frac{\sqrt{2}}{6})$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{π}{4} = 0 + 0 + 3 \sqrt{2} \cdot (x - \frac{\sqrt{2}}{6})$

Step 2.3.1.2

Simplify by adding zeros.

$y - \frac{π}{4} = 3 \sqrt{2} \cdot (x - \frac{\sqrt{2}}{6})$

Step 2.3.1.3

Apply the distributive property.

$y - \frac{π}{4} = 3 \sqrt{2} x + 3 \sqrt{2} (- \frac{\sqrt{2}}{6})$

Step 2.3.1.4

Cancel the common factor of $3$ .

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Step 2.3.1.4.1

Move the leading negative in $- \frac{\sqrt{2}}{6}$ into the numerator.

$y - \frac{π}{4} = 3 \sqrt{2} x + 3 \sqrt{2} \frac{- \sqrt{2}}{6}$

Step 2.3.1.4.2

Factor $3$ out of $3 \sqrt{2}$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + 3 (\sqrt{2}) \frac{- \sqrt{2}}{6}$

Step 2.3.1.4.3

Factor $3$ out of $6$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + 3 (\sqrt{2}) \frac{- \sqrt{2}}{3 (2)}$

Step 2.3.1.4.4

Cancel the common factor.

$y - \frac{π}{4} = 3 \sqrt{2} x + 3 \sqrt{2} \frac{- \sqrt{2}}{3 \cdot 2}$

Step 2.3.1.4.5

Rewrite the expression.

$y - \frac{π}{4} = 3 \sqrt{2} x + \sqrt{2} \frac{- \sqrt{2}}{2}$

Step 2.3.1.5

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

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{\sqrt{2} (- \sqrt{2})}{2}$

Step 2.3.1.6

Raise $\sqrt{2}$ to the power of $1$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- ({\sqrt{2}}^{1} \sqrt{2})}{2}$

Step 2.3.1.7

Raise $\sqrt{2}$ to the power of $1$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{2}$

Step 2.3.1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- {\sqrt{2}}^{1 + 1}}{2}$

Step 2.3.1.9

Add $1$ and $1$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- {\sqrt{2}}^{2}}{2}$

Step 2.3.1.10

Simplify each term.

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Step 2.3.1.10.1

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.3.1.10.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- {(2^{\frac{1}{2}})}^{2}}{2}$

Step 2.3.1.10.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 2^{\frac{1}{2} \cdot 2}}{2}$

Step 2.3.1.10.1.3

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

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 2^{\frac{2}{2}}}{2}$

Step 2.3.1.10.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.10.1.4.1

Cancel the common factor.

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 2^{\frac{2}{2}}}{2}$

Step 2.3.1.10.1.4.2

Rewrite the expression.

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 2^{1}}{2}$

Step 2.3.1.10.1.5

Evaluate the exponent.

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 1 \cdot 2}{2}$

Step 2.3.1.10.2

Multiply $- 1$ by $2$ .

$y - \frac{π}{4} = 3 \sqrt{2} x + \frac{- 2}{2}$

Step 2.3.1.10.3

Divide $- 2$ by $2$ .

$y - \frac{π}{4} = 3 \sqrt{2} x - 1$

Step 2.3.2

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

$y = 3 \sqrt{2} x - 1 + \frac{π}{4}$

Step 2.3.3

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

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Step 2.3.3.1

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

$y = 3 \sqrt{2} x - 1 \cdot \frac{4}{4} + \frac{π}{4}$

Step 2.3.3.2

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

$y = 3 \sqrt{2} x + \frac{- 1 \cdot 4}{4} + \frac{π}{4}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = 3 \sqrt{2} x + \frac{- 1 \cdot 4 + π}{4}$

Step 2.3.3.4

Multiply $- 1$ by $4$ .

$y = 3 \sqrt{2} x + \frac{- 4 + π}{4}$

Step 2.3.3.5

Rewrite $- 4$ as $- 1 (4)$ .

$y = 3 \sqrt{2} x + \frac{- 1 (4) + π}{4}$

Step 2.3.3.6

Factor $- 1$ out of $π$ .

$y = 3 \sqrt{2} x + \frac{- 1 (4) - 1 (- π)}{4}$

Step 2.3.3.7

Factor $- 1$ out of $- 1 (4) - 1 (- π)$ .

$y = 3 \sqrt{2} x + \frac{- 1 (4 - π)}{4}$

Step 2.3.3.8

Move the negative in front of the fraction.

$y = 3 \sqrt{2} x - \frac{4 - π}{4}$

Step 3