Calculus Examples

$\sin (x + y) = 2 x - 2 y$ , $(π, π)$

Step 1

Find the first derivative and evaluate at $x = π$ and $y = π$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (x + y)) = \frac{d}{d x} (2 x - 2 y)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.2

Differentiate.

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Step 1.2.2.1

By the Sum Rule, the derivative of $x + y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [y]$ .

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

Step 1.2.2.2

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

$\cos (x + y) (1 + \frac{d}{d x} [y])$

Step 1.2.3

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

$\cos (x + y) (1 + y')$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $2 x - 2 y$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 2 y]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [- 2 y]$

Step 1.3.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.3.2.1

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

$2 \frac{d}{d x} [x] + \frac{d}{d x} [- 2 y]$

Step 1.3.2.2

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

$2 \cdot 1 + \frac{d}{d x} [- 2 y]$

Step 1.3.2.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [- 2 y]$

Step 1.3.3

Evaluate $\frac{d}{d x} [- 2 y]$ .

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Step 1.3.3.1

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

$2 - 2 \frac{d}{d x} [y]$

Step 1.3.3.2

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

$2 - 2 y'$

Step 1.4

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

$\cos (x + y) (1 + y') = 2 - 2 y'$

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

Simplify $\cos (x + y) (1 + y')$ .

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Step 1.5.1.1.1

Rewrite.

$0 + 0 + \cos (x + y) (1 + y') = 2 - 2 y'$

Step 1.5.1.1.2

Simplify by adding zeros.

$\cos (x + y) (1 + y') = 2 - 2 y'$

Step 1.5.1.1.3

Apply the distributive property.

$\cos (x + y) \cdot 1 + \cos (x + y) y' = 2 - 2 y'$

Step 1.5.1.1.4

Simplify the expression.

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Step 1.5.1.1.4.1

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

$\cos (x + y) + \cos (x + y) y' = 2 - 2 y'$

Step 1.5.1.1.4.2

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

$\cos (x + y) + y' \cos (x + y) = 2 - 2 y'$

Step 1.5.2

Add $2 y'$ to both sides of the equation.

$\cos (x + y) + y' \cos (x + y) + 2 y' = 2$

Step 1.5.3

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

$y' \cos (x + y) + 2 y' = 2 - \cos (x + y)$

Step 1.5.4

Factor $y'$ out of $y' \cos (x + y) + 2 y'$ .

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Step 1.5.4.1

Factor $y'$ out of $y' \cos (x + y)$ .

$y' (\cos (x + y)) + 2 y' = 2 - \cos (x + y)$

Step 1.5.4.2

Factor $y'$ out of $2 y'$ .

$y' (\cos (x + y)) + y' \cdot 2 = 2 - \cos (x + y)$

Step 1.5.4.3

Factor $y'$ out of $y' (\cos (x + y)) + y' \cdot 2$ .

$y' (\cos (x + y) + 2) = 2 - \cos (x + y)$

Step 1.5.5

Divide each term in $y' (\cos (x + y) + 2) = 2 - \cos (x + y)$ by $\cos (x + y) + 2$ and simplify.

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Step 1.5.5.1

Divide each term in $y' (\cos (x + y) + 2) = 2 - \cos (x + y)$ by $\cos (x + y) + 2$ .

$\frac{y' (\cos (x + y) + 2)}{\cos (x + y) + 2} = \frac{2}{\cos (x + y) + 2} + \frac{- \cos (x + y)}{\cos (x + y) + 2}$

Step 1.5.5.2

Simplify the left side.

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Step 1.5.5.2.1

Cancel the common factor of $\cos (x + y) + 2$ .

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Step 1.5.5.2.1.1

Cancel the common factor.

$\frac{y' (\cos (x + y) + 2)}{\cos (x + y) + 2} = \frac{2}{\cos (x + y) + 2} + \frac{- \cos (x + y)}{\cos (x + y) + 2}$

Step 1.5.5.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2}{\cos (x + y) + 2} + \frac{- \cos (x + y)}{\cos (x + y) + 2}$

Step 1.5.5.3

Simplify the right side.

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Step 1.5.5.3.1

Move the negative in front of the fraction.

$y' = \frac{2}{\cos (x + y) + 2} - \frac{\cos (x + y)}{\cos (x + y) + 2}$

Step 1.5.5.3.2

Combine the numerators over the common denominator.

$y' = \frac{2 - \cos (x + y)}{\cos (x + y) + 2}$

Step 1.6

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

$\frac{d y}{d x} = \frac{2 - \cos (x + y)}{\cos (x + y) + 2}$

Step 1.7

Evaluate at $x = π$ and $y = π$ .

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Step 1.7.1

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

$\frac{2 - \cos ((π) + y)}{\cos ((π) + y) + 2}$

Step 1.7.2

Replace the variable $y$ with $π$ in the expression.

$\frac{2 - \cos (π + π)}{\cos (π + π) + 2}$

Step 1.7.3

Remove parentheses.

$\frac{2 - \cos (π + π)}{\cos (π + π) + 2}$

Step 1.7.4

Simplify the numerator.

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Step 1.7.4.1

Add $π$ and $π$ .

$\frac{2 - \cos (2 π)}{\cos (π + π) + 2}$

Step 1.7.4.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{2 - \cos (0)}{\cos (π + π) + 2}$

Step 1.7.4.3

The exact value of $\cos (0)$ is $1$ .

$\frac{2 - 1 \cdot 1}{\cos (π + π) + 2}$

Step 1.7.4.4

Multiply $- 1$ by $1$ .

$\frac{2 - 1}{\cos (π + π) + 2}$

Step 1.7.4.5

Subtract $1$ from $2$ .

$\frac{1}{\cos (π + π) + 2}$

Step 1.7.5

Simplify the denominator.

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Step 1.7.5.1

Add $π$ and $π$ .

$\frac{1}{\cos (2 π) + 2}$

Step 1.7.5.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{\cos (0) + 2}$

Step 1.7.5.3

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{1 + 2}$

Step 1.7.5.4

Add $1$ and $2$ .

$\frac{1}{3}$

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

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

Step 2.2

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

$y - π = \frac{1}{3} \cdot (x - π)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

$y - π = \frac{1}{3} \cdot (x - π)$

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Combine $\frac{1}{3}$ and $x$ .

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

Step 2.3.1.5

Combine $\frac{1}{3}$ and $π$ .

$y - π = \frac{x}{3} - \frac{π}{3}$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $π$ to both sides of the equation.

$y = \frac{x}{3} - \frac{π}{3} + π$

Step 2.3.2.2

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

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

Step 2.3.2.3

Combine $π$ and $\frac{3}{3}$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Combine the numerators over the common denominator.

$y = \frac{x - π + π \cdot 3}{3}$

Step 2.3.2.6

Move $3$ to the left of $π$ .

$y = \frac{x - π + 3 π}{3}$

Step 2.3.2.7

Add $- π$ and $3 π$ .

$y = \frac{x + 2 π}{3}$

Step 2.3.2.8

Split the fraction $\frac{x + 2 π}{3}$ into two fractions.

$y = \frac{x}{3} + \frac{2 π}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{3} x + \frac{2 π}{3}$

Step 3