Calculus Examples

$x^{2} - y^{2} - 3 x - 19 = 0$ , $(- 4, 3)$

Step 1

Find the first derivative and evaluate at $x = - 4$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} - y^{2} - 3 x - 19) = \frac{d}{d x} (0)$

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 $x^{2} - y^{2} - 3 x - 19$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

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 $n = 2$ .

$2 x + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2

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

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Step 1.2.2.1

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

$2 x - \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2.2

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

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Step 1.2.2.2.1

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

$2 x - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2.2.2

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

$2 x - (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2.2.3

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

$2 x - (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2.3

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

$2 x - (2 y y') + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.2.4

Multiply $2$ by $- 1$ .

$2 x - 2 y y' + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [- 19]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.2.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$2 x - 2 y y' - 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 19]$

Step 1.2.3.2

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

$2 x - 2 y y' - 3 \cdot 1 + \frac{d}{d x} [- 19]$

Step 1.2.3.3

Multiply $- 3$ by $1$ .

$2 x - 2 y y' - 3 + \frac{d}{d x} [- 19]$

Step 1.2.4

Since $- 19$ is constant with respect to $x$ , the derivative of $- 19$ with respect to $x$ is $0$ .

$2 x - 2 y y' - 3 + 0$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Add $2 x - 2 y y' - 3$ and $0$ .

$2 x - 2 y y' - 3$

Step 1.2.5.2

Reorder terms.

$- 2 y y' + 2 x - 3$

Step 1.3

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

$0$

Step 1.4

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

$- 2 y y' + 2 x - 3 = 0$

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.

$- 2 y y' - 3 = - 2 x$

Step 1.5.1.2

Add $3$ to both sides of the equation.

$- 2 y y' = - 2 x + 3$

Step 1.5.2

Divide each term in $- 2 y y' = - 2 x + 3$ by $- 2 y$ and simplify.

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Step 1.5.2.1

Divide each term in $- 2 y y' = - 2 x + 3$ by $- 2 y$ .

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Rewrite the expression.

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

Step 1.5.2.2.2

Cancel the common factor of $y$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Cancel the common factor of $- 2$ .

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Step 1.5.2.3.1.1.1

Cancel the common factor.

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

Step 1.5.2.3.1.1.2

Rewrite the expression.

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

Step 1.5.2.3.1.2

Move the negative in front of the fraction.

$y' = \frac{x}{y} - \frac{3}{2 y}$

Step 1.6

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

$\frac{d y}{d x} = \frac{x}{y} - \frac{3}{2 y}$

Step 1.7

Evaluate at $x = - 4$ and $y = 3$ .

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Step 1.7.1

Replace the variable $x$ with $- 4$ in the expression.

$\frac{- 4}{y} - \frac{3}{2 y}$

Step 1.7.2

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

$\frac{- 4}{3} - \frac{3}{2 (3)}$

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Move the negative in front of the fraction.

$- \frac{4}{3} - \frac{3}{2 (3)}$

Step 1.7.3.2

Cancel the common factor of $3$ .

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Step 1.7.3.2.1

Cancel the common factor.

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

Step 1.7.3.2.2

Rewrite the expression.

$- \frac{4}{3} - \frac{1}{2}$

Step 1.7.4

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{4}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 1.7.5

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{4}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 1.7.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.7.6.1

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$- \frac{4 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 1.7.6.2

Multiply $3$ by $2$ .

$- \frac{4 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 1.7.6.3

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

$- \frac{4 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 1.7.6.4

Multiply $2$ by $3$ .

$- \frac{4 \cdot 2}{6} - \frac{3}{6}$

Step 1.7.7

Combine the numerators over the common denominator.

$\frac{- 4 \cdot 2 - 3}{6}$

Step 1.7.8

Simplify the numerator.

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Step 1.7.8.1

Multiply $- 4$ by $2$ .

$\frac{- 8 - 3}{6}$

Step 1.7.8.2

Subtract $3$ from $- 8$ .

$\frac{- 11}{6}$

Step 1.7.9

Move the negative in front of the fraction.

$- \frac{11}{6}$

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 $- \frac{11}{6}$ and a given point $(- 4, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = - \frac{11}{6} \cdot (x - (- 4))$

Step 2.2

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

$y - 3 = - \frac{11}{6} \cdot (x + 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{11}{6} \cdot (x + 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 3 = 0 + 0 - \frac{11}{6} \cdot (x + 4)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 3 = - \frac{11}{6} x - \frac{11}{6} \cdot 4$

Step 2.3.1.2.2

Combine $x$ and $\frac{11}{6}$ .

$y - 3 = - \frac{x \cdot 11}{6} - \frac{11}{6} \cdot 4$

Step 2.3.1.2.3

Cancel the common factor of $2$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{11}{6}$ into the numerator.

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11}{6} \cdot 4$

Step 2.3.1.2.3.2

Factor $2$ out of $6$ .

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11}{2 (3)} \cdot 4$

Step 2.3.1.2.3.3

Factor $2$ out of $4$ .

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11}{2 \cdot 3} \cdot (2 \cdot 2)$

Step 2.3.1.2.3.4

Cancel the common factor.

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11}{2 \cdot 3} \cdot (2 \cdot 2)$

Step 2.3.1.2.3.5

Rewrite the expression.

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11}{3} \cdot 2$

Step 2.3.1.2.4

Combine $\frac{- 11}{3}$ and $2$ .

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 11 \cdot 2}{3}$

Step 2.3.1.2.5

Multiply $- 11$ by $2$ .

$y - 3 = - \frac{x \cdot 11}{6} + \frac{- 22}{3}$

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Move $11$ to the left of $x$ .

$y - 3 = - \frac{11 \cdot x}{6} + \frac{- 22}{3}$

Step 2.3.1.3.2

Move the negative in front of the fraction.

$y - 3 = - \frac{11 x}{6} - \frac{22}{3}$

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 $3$ to both sides of the equation.

$y = - \frac{11 x}{6} - \frac{22}{3} + 3$

Step 2.3.2.2

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

$y = - \frac{11 x}{6} - \frac{22}{3} + 3 \cdot \frac{3}{3}$

Step 2.3.2.3

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

$y = - \frac{11 x}{6} - \frac{22}{3} + \frac{3 \cdot 3}{3}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{11 x}{6} + \frac{- 22 + 3 \cdot 3}{3}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $3$ .

$y = - \frac{11 x}{6} + \frac{- 22 + 9}{3}$

Step 2.3.2.5.2

Add $- 22$ and $9$ .

$y = - \frac{11 x}{6} + \frac{- 13}{3}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = - \frac{11 x}{6} - \frac{13}{3}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{11}{6} x) - \frac{13}{3}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{11}{6} x - \frac{13}{3}$

Step 3