Calculus Examples

$x^{2} - x y - y^{2} = 1$ , $(5, 3)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

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 $x^{2} - x y - y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x y] + \frac{d}{d x} [- y^{2}]$ .

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

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 $n = 2$ .

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

Step 1.2.2

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

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Step 1.2.2.1

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

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

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 $g (x) = y$ .

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

Step 1.2.2.3

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

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

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 $n = 1$ .

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

Step 1.2.2.5

Multiply $y$ by $1$ .

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

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

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

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 $y$ .

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

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 $n = 2$ .

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

Step 1.2.3.2.3

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Multiply $2$ by $- 1$ .

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

Apply the distributive property.

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

Step 1.2.4.2

Remove unnecessary parentheses.

$2 x - x y' - y - 2 y y'$

Step 1.2.4.3

Reorder terms.

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

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.

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

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.

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

Step 1.5.1.2

Add $y$ to both sides of the equation.

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.2.3

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

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Simplify the left side.

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Step 1.5.3.2.1

Cancel the common factor of $- x - 2 y$ .

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Step 1.5.3.2.1.1

Cancel the common factor.

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

Step 1.5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 1.5.3.3

Simplify the right side.

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Step 1.5.3.3.1

Combine the numerators over the common denominator.

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

Step 1.5.3.3.2

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

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

Step 1.5.3.3.3

Factor $- 1$ out of $y$ .

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

Step 1.5.3.3.4

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

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

Step 1.5.3.3.5

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

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

Step 1.5.3.3.6

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

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

Step 1.5.3.3.7

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

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

Step 1.5.3.3.8

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

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

Step 1.5.3.3.9

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

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

Step 1.5.3.3.10

Cancel the common factor.

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

Step 1.5.3.3.11

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 5$ and $y = 3$ .

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Step 1.7.1

Replace the variable $x$ with $5$ in the expression.

$\frac{2 (5) - y}{(5) + 2 y}$

Step 1.7.2

Replace the variable $y$ with $3$ in the expression.

$\frac{2 (5) - (3)}{(5) + 2 (3)}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Multiply $2$ by $5$ .

$\frac{10 - (3)}{5 + 2 (3)}$

Step 1.7.3.2

Multiply $- 1$ by $3$ .

$\frac{10 - 3}{5 + 2 (3)}$

Step 1.7.3.3

Subtract $3$ from $10$ .

$\frac{7}{5 + 2 (3)}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $2$ by $3$ .

$\frac{7}{5 + 6}$

Step 1.7.4.2

Add $5$ and $6$ .

$\frac{7}{11}$

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 $\frac{7}{11}$ and a given point $(5, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{7}{11} \cdot (x - (5))$

Step 2.2

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

$y - 3 = \frac{7}{11} \cdot (x - 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{7}{11} \cdot (x - 5)$ .

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Step 2.3.1.1

Rewrite.

$y - 3 = 0 + 0 + \frac{7}{11} \cdot (x - 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 3 = \frac{7}{11} \cdot (x - 5)$

Step 2.3.1.3

Apply the distributive property.

$y - 3 = \frac{7}{11} x + \frac{7}{11} \cdot - 5$

Step 2.3.1.4

Combine $\frac{7}{11}$ and $x$ .

$y - 3 = \frac{7 x}{11} + \frac{7}{11} \cdot - 5$

Step 2.3.1.5

Multiply $\frac{7}{11} \cdot - 5$ .

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Step 2.3.1.5.1

Combine $\frac{7}{11}$ and $- 5$ .

$y - 3 = \frac{7 x}{11} + \frac{7 \cdot - 5}{11}$

Step 2.3.1.5.2

Multiply $7$ by $- 5$ .

$y - 3 = \frac{7 x}{11} + \frac{- 35}{11}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 3 = \frac{7 x}{11} - \frac{35}{11}$

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 $3$ to both sides of the equation.

$y = \frac{7 x}{11} - \frac{35}{11} + 3$

Step 2.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$y = \frac{7 x}{11} - \frac{35}{11} + 3 \cdot \frac{11}{11}$

Step 2.3.2.3

Combine $3$ and $\frac{11}{11}$ .

$y = \frac{7 x}{11} - \frac{35}{11} + \frac{3 \cdot 11}{11}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{7 x}{11} + \frac{- 35 + 3 \cdot 11}{11}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $11$ .

$y = \frac{7 x}{11} + \frac{- 35 + 33}{11}$

Step 2.3.2.5.2

Add $- 35$ and $33$ .

$y = \frac{7 x}{11} + \frac{- 2}{11}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{7 x}{11} - \frac{2}{11}$

Step 2.3.3

Reorder terms.

$y = \frac{7}{11} x - \frac{2}{11}$

Step 3