Calculus Examples

$x^{2} + 2 x y - y^{2} + x = 6$ , $(2, 4)$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 4$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} + 2 x y - y^{2} + x) = \frac{d}{d x} (6)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

Step 1.2.2.2

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 $g (x) = y$ .

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

Step 1.2.2.3

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

$2 x + 2 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [x]$

Step 1.2.2.4

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

$2 x + 2 (x y' + y \cdot 1) + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [x]$

Step 1.2.2.5

Multiply $y$ by $1$ .

$2 x + 2 (x y' + y) + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [x]$

Step 1.2.3

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

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Step 1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

$2 x + 2 (x y' + y) - \frac{d}{d x} [y^{2}] + \frac{d}{d x} [x]$

Step 1.2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.3.2.1

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

$2 x + 2 (x y' + y) - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [x]$

Step 1.2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 x + 2 (x y' + y) - (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [x]$

Step 1.2.3.2.3

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Multiply $2$ by $- 1$ .

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

Step 1.2.4

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

$2 x + 2 (x y' + y) - 2 y y' + 1$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Apply the distributive property.

$2 x + 2 (x y') + 2 y - 2 y y' + 1$

Step 1.2.5.2

Remove unnecessary parentheses.

$2 x + 2 x y' + 2 y - 2 y y' + 1$

Step 1.2.5.3

Reorder terms.

$2 x y' - 2 y y' + 2 x + 2 y + 1$

Step 1.3

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$2 x y' - 2 y y' + 2 x + 2 y + 1 = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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Step 1.5.1.1

Subtract $2 x$ from both sides of the equation.

$2 x y' - 2 y y' + 2 y + 1 = - 2 x$

Step 1.5.1.2

Subtract $2 y$ from both sides of the equation.

$2 x y' - 2 y y' + 1 = - 2 x - 2 y$

Step 1.5.1.3

Subtract $1$ from both sides of the equation.

$2 x y' - 2 y y' = - 2 x - 2 y - 1$

Step 1.5.2

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

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Step 1.5.2.1

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

$2 y' x - 2 y y' = - 2 x - 2 y - 1$

Step 1.5.2.2

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

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

Step 1.5.2.3

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

$2 y' (x - y) = - 2 x - 2 y - 1$

Step 1.5.3

Divide each term in $2 y' (x - y) = - 2 x - 2 y - 1$ by $2 (x - y)$ and simplify.

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Step 1.5.3.1

Divide each term in $2 y' (x - y) = - 2 x - 2 y - 1$ by $2 (x - y)$ .

$\frac{2 y' (x - y)}{2 (x - y)} = \frac{- 2 x}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $2$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

$\frac{2 y' (x - y)}{2 (x - y)} = \frac{- 2 x}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.2.1.2

Rewrite the expression.

$\frac{y' (x - y)}{x - y} = \frac{- 2 x}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.2.2

Cancel the common factor of $x - y$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

$\frac{y' (x - y)}{x - y} = \frac{- 2 x}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Simplify each term.

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Step 1.5.3.3.1.1

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

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Step 1.5.3.3.1.1.1

Factor $2$ out of $- 2 x$ .

$y' = \frac{2 (- x)}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.1.2

Cancel the common factors.

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Step 1.5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{2 (- x)}{2 (x - y)} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- x}{x - y} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{x}{x - y} + \frac{- 2 y}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.3

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

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Step 1.5.3.3.1.3.1

Factor $2$ out of $- 2 y$ .

$y' = - \frac{x}{x - y} + \frac{2 (- y)}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.3.2

Cancel the common factors.

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Step 1.5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{x}{x - y} + \frac{2 (- y)}{2 (x - y)} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{x}{x - y} + \frac{- y}{x - y} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{x}{x - y} - \frac{y}{x - y} + \frac{- 1}{2 (x - y)}$

Step 1.5.3.3.1.5

Move the negative in front of the fraction.

$y' = - \frac{x}{x - y} - \frac{y}{x - y} - \frac{1}{2 (x - y)}$

Step 1.5.3.3.2

Combine the numerators over the common denominator.

$y' = \frac{- x - y}{x - y} - \frac{1}{2 (x - y)}$

Step 1.5.3.3.3

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

$y' = \frac{- x - y}{x - y} \cdot \frac{2}{2} - \frac{1}{2 (x - y)}$

Step 1.5.3.3.4

Write each expression with a common denominator of $(x - y) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.3.4.1

Multiply $\frac{- x - y}{x - y}$ by $\frac{2}{2}$ .

$y' = \frac{(- x - y) \cdot 2}{(x - y) \cdot 2} - \frac{1}{2 (x - y)}$

Step 1.5.3.3.4.2

Reorder the factors of $(x - y) \cdot 2$ .

$y' = \frac{(- x - y) \cdot 2}{2 (x - y)} - \frac{1}{2 (x - y)}$

Step 1.5.3.3.5

Combine the numerators over the common denominator.

$y' = \frac{(- x - y) \cdot 2 - 1}{2 (x - y)}$

Step 1.5.3.3.6

Simplify the numerator.

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Step 1.5.3.3.6.1

Apply the distributive property.

$y' = \frac{- x \cdot 2 - y \cdot 2 - 1}{2 (x - y)}$

Step 1.5.3.3.6.2

Multiply $2$ by $- 1$ .

$y' = \frac{- 2 x - y \cdot 2 - 1}{2 (x - y)}$

Step 1.5.3.3.6.3

Multiply $2$ by $- 1$ .

$y' = \frac{- 2 x - 2 y - 1}{2 (x - y)}$

Step 1.5.3.3.7

Simplify with factoring out.

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Step 1.5.3.3.7.1

Factor $- 1$ out of $- 2 x$ .

$y' = \frac{- (2 x) - 2 y - 1}{2 (x - y)}$

Step 1.5.3.3.7.2

Factor $- 1$ out of $- 2 y$ .

$y' = \frac{- (2 x) - (2 y) - 1}{2 (x - y)}$

Step 1.5.3.3.7.3

Factor $- 1$ out of $- (2 x) - (2 y)$ .

$y' = \frac{- (2 x + 2 y) - 1}{2 (x - y)}$

Step 1.5.3.3.7.4

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

$y' = \frac{- (2 x + 2 y) - 1 (1)}{2 (x - y)}$

Step 1.5.3.3.7.5

Factor $- 1$ out of $- (2 x + 2 y) - 1 (1)$ .

$y' = \frac{- (2 x + 2 y + 1)}{2 (x - y)}$

Step 1.5.3.3.7.6

Simplify the expression.

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Step 1.5.3.3.7.6.1

Rewrite $- (2 x + 2 y + 1)$ as $- 1 (2 x + 2 y + 1)$ .

$y' = \frac{- 1 (2 x + 2 y + 1)}{2 (x - y)}$

Step 1.5.3.3.7.6.2

Move the negative in front of the fraction.

$y' = - \frac{2 x + 2 y + 1}{2 (x - y)}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{2 x + 2 y + 1}{2 (x - y)}$

Step 1.7

Evaluate at $x = 2$ and $y = 4$ .

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Step 1.7.1

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

$- \frac{2 (2) + 2 y + 1}{2 ((2) - y)}$

Step 1.7.2

Replace the variable $y$ with $4$ in the expression.

$- \frac{2 (2) + 2 (4) + 1}{2 ((2) - (4))}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Multiply $2$ by $2$ .

$- \frac{4 + 2 (4) + 1}{2 (2 - (4))}$

Step 1.7.3.2

Multiply $2$ by $4$ .

$- \frac{4 + 8 + 1}{2 (2 - (4))}$

Step 1.7.3.3

Add $4$ and $8$ .

$- \frac{12 + 1}{2 (2 - (4))}$

Step 1.7.3.4

Add $12$ and $1$ .

$- \frac{13}{2 (2 - (4))}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $- 1$ by $4$ .

$- \frac{13}{2 (2 - 4)}$

Step 1.7.4.2

Subtract $4$ from $2$ .

$- \frac{13}{2 \cdot - 2}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Multiply $2$ by $- 2$ .

$- \frac{13}{- 4}$

Step 1.7.5.2

Move the negative in front of the fraction.

$- - \frac{13}{4}$

Step 1.7.6

Multiply $- - \frac{13}{4}$ .

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Step 1.7.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{13}{4})$

Step 1.7.6.2

Multiply $\frac{13}{4}$ by $1$ .

$\frac{13}{4}$

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 $\frac{13}{4}$ and a given point $(2, 4)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = \frac{13}{4} \cdot (x - (2))$

Step 2.2

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

$y - 4 = \frac{13}{4} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{13}{4} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{13}{4} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = \frac{13}{4} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = \frac{13}{4} x + \frac{13}{4} \cdot - 2$

Step 2.3.1.4

Combine $\frac{13}{4}$ and $x$ .

$y - 4 = \frac{13 x}{4} + \frac{13}{4} \cdot - 2$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $4$ .

$y - 4 = \frac{13 x}{4} + \frac{13}{2 (2)} \cdot - 2$

Step 2.3.1.5.2

Factor $2$ out of $- 2$ .

$y - 4 = \frac{13 x}{4} + \frac{13}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 4 = \frac{13 x}{4} + \frac{13}{2 \cdot 2} \cdot (2 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 4 = \frac{13 x}{4} + \frac{13}{2} \cdot - 1$

Step 2.3.1.6

Combine $\frac{13}{2}$ and $- 1$ .

$y - 4 = \frac{13 x}{4} + \frac{13 \cdot - 1}{2}$

Step 2.3.1.7

Simplify the expression.

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Step 2.3.1.7.1

Multiply $13$ by $- 1$ .

$y - 4 = \frac{13 x}{4} + \frac{- 13}{2}$

Step 2.3.1.7.2

Move the negative in front of the fraction.

$y - 4 = \frac{13 x}{4} - \frac{13}{2}$

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 $4$ to both sides of the equation.

$y = \frac{13 x}{4} - \frac{13}{2} + 4$

Step 2.3.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{13 x}{4} - \frac{13}{2} + 4 \cdot \frac{2}{2}$

Step 2.3.2.3

Combine $4$ and $\frac{2}{2}$ .

$y = \frac{13 x}{4} - \frac{13}{2} + \frac{4 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{13 x}{4} + \frac{- 13 + 4 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $4$ by $2$ .

$y = \frac{13 x}{4} + \frac{- 13 + 8}{2}$

Step 2.3.2.5.2

Add $- 13$ and $8$ .

$y = \frac{13 x}{4} + \frac{- 5}{2}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{13 x}{4} - \frac{5}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{13}{4} x - \frac{5}{2}$

Step 3