Calculus Examples

$x \cos (y) = 1$ , $(2, \frac{π}{3})$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = \frac{π}{3}$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x \cos (y)) = \frac{d}{d x} (1)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \cos (y)$ .

$x \frac{d}{d x} [\cos (y)] + \cos (y) \frac{d}{d x} [x]$

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = y$ .

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $y$ .

$x (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [x]$

Step 1.2.2.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$x (- \sin (u) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [x]$

Step 1.2.2.3

Replace all occurrences of $u$ with $y$ .

$x (- \sin (y) \frac{d}{d x} [y]) + \cos (y) \frac{d}{d x} [x]$

Step 1.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$x (- \sin (y) y') + \cos (y) \frac{d}{d x} [x]$

Step 1.2.4

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

$x (- \sin (y) y') + \cos (y) \cdot 1$

Step 1.2.5

Simplify the expression.

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Step 1.2.5.1

Multiply $\cos (y)$ by $1$ .

$x (- \sin (y) y') + \cos (y)$

Step 1.2.5.2

Reorder terms.

$- x \sin (y) y' + \cos (y)$

Step 1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$- x \sin (y) y' + \cos (y) = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Reorder factors in $- x \sin (y) y' + \cos (y)$ .

$- x y' \sin (y) + \cos (y) = 0$

Step 1.5.2

Subtract $\cos (y)$ from both sides of the equation.

$- x y' \sin (y) = - \cos (y)$

Step 1.5.3

Divide each term in $- x y' \sin (y) = - \cos (y)$ by $- x \sin (y)$ and simplify.

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Step 1.5.3.1

Divide each term in $- x y' \sin (y) = - \cos (y)$ by $- x \sin (y)$ .

$\frac{- x y' \sin (y)}{- x \sin (y)} = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Dividing two negative values results in a positive value.

$\frac{x y' \sin (y)}{x \sin (y)} = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.2.2

Cancel the common factor of $x$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

$\frac{x y' \sin (y)}{x \sin (y)} = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.2.2.2

Rewrite the expression.

$\frac{y' \sin (y)}{\sin (y)} = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.2.3

Cancel the common factor of $\sin (y)$ .

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Step 1.5.3.2.3.1

Cancel the common factor.

$\frac{y' \sin (y)}{\sin (y)} = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- \cos (y)}{- x \sin (y)}$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Dividing two negative values results in a positive value.

$y' = \frac{\cos (y)}{x \sin (y)}$

Step 1.5.3.3.2

Separate fractions.

$y' = \frac{1}{x} \cdot \frac{\cos (y)}{\sin (y)}$

Step 1.5.3.3.3

Convert from $\frac{\cos (y)}{\sin (y)}$ to $\cot (y)$ .

$y' = \frac{1}{x} \cot (y)$

Step 1.5.3.3.4

Combine $\frac{1}{x}$ and $\cot (y)$ .

$y' = \frac{\cot (y)}{x}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{\cot (y)}{x}$

Step 1.7

Evaluate at $x = 2$ and $y = \frac{π}{3}$ .

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Step 1.7.1

Replace the variable $x$ with $2$ in the expression.

$\frac{\cot (y)}{2}$

Step 1.7.2

Replace the variable $y$ with $\frac{π}{3}$ in the expression.

$\frac{\cot (\frac{π}{3})}{2}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

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

Step 1.7.3.2

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.7.3.3

Combine and simplify the denominator.

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Step 1.7.3.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.7.3.3.2

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

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

Step 1.7.3.3.3

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

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

Step 1.7.3.3.4

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

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

Step 1.7.3.3.5

Add $1$ and $1$ .

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

Step 1.7.3.3.6

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

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Step 1.7.3.3.6.1

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

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

Step 1.7.3.3.6.2

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

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

Step 1.7.3.3.6.3

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

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

Step 1.7.3.3.6.4

Cancel the common factor of $2$ .

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Step 1.7.3.3.6.4.1

Cancel the common factor.

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

Step 1.7.3.3.6.4.2

Rewrite the expression.

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

Step 1.7.3.3.6.5

Evaluate the exponent.

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

Step 1.7.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.7.5

Multiply $\frac{\sqrt{3}}{3} \cdot \frac{1}{2}$ .

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Step 1.7.5.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{1}{2}$ .

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

Step 1.7.5.2

Multiply $3$ by $2$ .

$\frac{\sqrt{3}}{6}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Combine $\frac{\sqrt{3}}{6}$ and $x$ .

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

Step 2.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.3.1.2.3.1

Factor $2$ out of $6$ .

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

Step 2.3.1.2.3.2

Factor $2$ out of $- 2$ .

$y - \frac{π}{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{2 \cdot 3} \cdot (2 \cdot - 1)$

Step 2.3.1.2.3.3

Cancel the common factor.

$y - \frac{π}{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{2 \cdot 3} \cdot (2 \cdot - 1)$

Step 2.3.1.2.3.4

Rewrite the expression.

$y - \frac{π}{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3}}{3} \cdot - 1$

Step 2.3.1.2.4

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

$y - \frac{π}{3} = \frac{\sqrt{3} x}{6} + \frac{\sqrt{3} \cdot - 1}{3}$

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Simplify the numerator.

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Step 2.3.1.3.1.1

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

$y - \frac{π}{3} = \frac{\sqrt{3} x}{6} + \frac{- 1 \cdot \sqrt{3}}{3}$

Step 2.3.1.3.1.2

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

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

Step 2.3.1.3.2

Move the negative in front of the fraction.

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

Step 2.3.2

Add $\frac{π}{3}$ to both sides of the equation.

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

Step 2.3.3

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

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Step 2.3.3.1

Combine the numerators over the common denominator.

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

Step 2.3.3.2

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

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

Step 2.3.3.3

Factor $- 1$ out of $π$ .

$y = \frac{\sqrt{3} x}{6} + \frac{- (\sqrt{3}) - 1 (- π)}{3}$

Step 2.3.3.4

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

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

Step 2.3.3.5

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

$y = \frac{\sqrt{3} x}{6} + \frac{- 1 (\sqrt{3} - π)}{3}$

Step 2.3.3.6

Move the negative in front of the fraction.

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

Step 2.3.3.7

Reorder terms.

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

Step 3