Calculus Examples

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

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.

$\frac{d}{d x} (x^{2} + x y - y^{2}) = \frac{d}{d x} (- 4)$

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

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 1.2.2.2

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 1.2.2.3

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 1.2.2.4

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

Reorder terms.

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

Step 1.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ 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

Subtract $y$ from 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 $- y$ .

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

Step 1.5.3.3.4

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

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

Step 1.5.3.3.5

Simplify the expression.

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Step 1.5.3.3.5.1

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

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

Step 1.5.3.3.5.2

Move the negative in front of the fraction.

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

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Reduce the expression by cancelling the common factors.

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Step 1.7.3.1

Remove parentheses.

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

Step 1.7.3.2

Cancel the common factor of $2 (2) + 4$ and $(2) - 2 \cdot 4$ .

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Step 1.7.3.2.1

Factor $2$ out of $4$ .

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

Step 1.7.3.2.2

Factor $2$ out of $2 (2) + 2 (2)$ .

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

Step 1.7.3.2.3

Cancel the common factors.

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Step 1.7.3.2.3.1

Factor $2$ out of $2$ .

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

Step 1.7.3.2.3.2

Factor $2$ out of $- 2 \cdot 4$ .

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

Step 1.7.3.2.3.3

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 4)$ .

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

Step 1.7.3.2.3.4

Cancel the common factor.

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

Step 1.7.3.2.3.5

Rewrite the expression.

$- \frac{2 + 2}{1 - 1 \cdot 4}$

Step 1.7.3.3

Add $2$ and $2$ .

$- \frac{4}{1 - 1 \cdot 4}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $- 1$ by $4$ .

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

Step 1.7.4.2

Subtract $4$ from $1$ .

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

Step 1.7.5

Move the negative in front of the fraction.

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

Step 1.7.6

Multiply $- - \frac{4}{3}$ .

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Step 1.7.6.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{4}{3})$

Step 1.7.6.2

Multiply $\frac{4}{3}$ by $1$ .

$\frac{4}{3}$

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 $\frac{4}{3}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4) = \frac{4}{3} \cdot (x - (2))$

Step 2.2

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

$y - 4 = \frac{4}{3} \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{4}{3} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 + \frac{4}{3} \cdot (x - 2)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = \frac{4}{3} \cdot (x - 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = \frac{4}{3} x + \frac{4}{3} \cdot - 2$

Step 2.3.1.4

Combine $\frac{4}{3}$ and $x$ .

$y - 4 = \frac{4 x}{3} + \frac{4}{3} \cdot - 2$

Step 2.3.1.5

Multiply $\frac{4}{3} \cdot - 2$ .

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Step 2.3.1.5.1

Combine $\frac{4}{3}$ and $- 2$ .

$y - 4 = \frac{4 x}{3} + \frac{4 \cdot - 2}{3}$

Step 2.3.1.5.2

Multiply $4$ by $- 2$ .

$y - 4 = \frac{4 x}{3} + \frac{- 8}{3}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 4 = \frac{4 x}{3} - \frac{8}{3}$

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 $4$ to both sides of the equation.

$y = \frac{4 x}{3} - \frac{8}{3} + 4$

Step 2.3.2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{4 x}{3} - \frac{8}{3} + 4 \cdot \frac{3}{3}$

Step 2.3.2.3

Combine $4$ and $\frac{3}{3}$ .

$y = \frac{4 x}{3} - \frac{8}{3} + \frac{4 \cdot 3}{3}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{4 x}{3} + \frac{- 8 + 4 \cdot 3}{3}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $4$ by $3$ .

$y = \frac{4 x}{3} + \frac{- 8 + 12}{3}$

Step 2.3.2.5.2

Add $- 8$ and $12$ .

$y = \frac{4 x}{3} + \frac{4}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{4}{3} x + \frac{4}{3}$

Step 3