Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = \frac{1}{2}$ and $y = \frac{π}{2}$ to find the slope of the tangent line.

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Step 1.1

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

$3 \frac{d}{d x} [\arcsin (x)]$

Step 1.2

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

$3 \frac{1}{\sqrt{1 - x^{2}}}$

Step 1.3

Combine $3$ and $\frac{1}{\sqrt{1 - x^{2}}}$ .

$\frac{3}{\sqrt{1 - x^{2}}}$

Step 1.4

Evaluate the derivative at $x = \frac{1}{2}$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Simplify the denominator.

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Step 1.5.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.5.1.2

One to any power is one.

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

Step 1.5.1.3

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

$\frac{3}{\sqrt{1 - \frac{1}{4}}}$

Step 1.5.1.4

Write $1$ as a fraction with a common denominator.

$\frac{3}{\sqrt{\frac{4}{4} - \frac{1}{4}}}$

Step 1.5.1.5

Combine the numerators over the common denominator.

$\frac{3}{\sqrt{\frac{4 - 1}{4}}}$

Step 1.5.1.6

Subtract $1$ from $4$ .

$\frac{3}{\sqrt{\frac{3}{4}}}$

Step 1.5.1.7

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

$\frac{3}{\frac{\sqrt{3}}{\sqrt{4}}}$

Step 1.5.1.8

Simplify the denominator.

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Step 1.5.1.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.5.1.8.2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 1.5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.3

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.5.4

Combine and simplify the denominator.

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Step 1.5.4.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 1.5.4.4

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

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

Step 1.5.4.5

Add $1$ and $1$ .

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

Step 1.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.5.4.6.2

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

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

Step 1.5.4.6.3

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

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

Step 1.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.5.4.6.4.1

Cancel the common factor.

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

Step 1.5.4.6.4.2

Rewrite the expression.

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

Step 1.5.4.6.5

Evaluate the exponent.

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

Step 1.5.5

Cancel the common factor of $3$ .

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Step 1.5.5.1

Cancel the common factor.

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

Step 1.5.5.2

Rewrite the expression.

$2 \sqrt{3}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $2 \sqrt{3} \cdot (x - \frac{1}{2})$ .

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Step 2.3.1.1

Rewrite.

$y - \frac{π}{2} = 0 + 0 + 2 \sqrt{3} \cdot (x - \frac{1}{2})$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Cancel the common factor of $2$ .

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Step 2.3.1.2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.3.1.2.2.2

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

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

Step 2.3.1.2.2.3

Cancel the common factor.

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

Step 2.3.1.2.2.4

Rewrite the expression.

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

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 2.3.1.3.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 2.3.2

Add $\frac{π}{2}$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

To write $- \sqrt{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.3.2

Combine $- \sqrt{3}$ and $\frac{2}{2}$ .

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

Step 2.3.3.3

Combine the numerators over the common denominator.

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

Step 2.3.3.4

Multiply $2$ by $- 1$ .

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

Step 2.3.3.5

Factor $- 1$ out of $- 2 \sqrt{3}$ .

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

Step 2.3.3.6

Factor $- 1$ out of $π$ .

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

Step 2.3.3.7

Factor $- 1$ out of $- (2 \sqrt{3}) - 1 (- π)$ .

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

Step 2.3.3.8

Rewrite $- (2 \sqrt{3} - π)$ as $- 1 (2 \sqrt{3} - π)$ .

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

Step 2.3.3.9

Move the negative in front of the fraction.

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

Step 3