Calculus Examples

${(x + 2)}^{2} + {(y - 3)}^{2} = 37$ , $(4, 4)$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x + 2)}^{2} + {(y - 3)}^{2}) = \frac{d}{d x} (37)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

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

Step 1.2.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Apply the distributive property.

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

Step 1.2.2.3

Apply the distributive property.

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

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 1.2.3.1.2

Move $2$ to the left of $x$ .

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

Step 1.2.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [x^{2} + 2 x + 2 x + 4 + {(y - 3)}^{2}]$

Step 1.2.3.2

Add $2 x$ and $2 x$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + {(y - 3)}^{2}]$

Step 1.2.4

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + (y - 3) (y - 3)]$

Step 1.2.5

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 1.2.5.1

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 4 x + 4 + y (y - 3) - 3 (y - 3)]$

Step 1.2.5.2

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 4 x + 4 + y \cdot y + y \cdot - 3 - 3 (y - 3)]$

Step 1.2.5.3

Apply the distributive property.

$\frac{d}{d x} [x^{2} + 4 x + 4 + y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3]$

Step 1.2.6

Simplify and combine like terms.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Multiply $y$ by $y$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3]$

Step 1.2.6.1.2

Move $- 3$ to the left of $y$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3]$

Step 1.2.6.1.3

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + y^{2} - 3 y - 3 y + 9]$

Step 1.2.6.2

Subtract $3 y$ from $- 3 y$ .

$\frac{d}{d x} [x^{2} + 4 x + 4 + y^{2} - 6 y + 9]$

Step 1.2.7

Differentiate.

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Step 1.2.7.1

By the Sum Rule, the derivative of $x^{2} + 4 x + 4 + y^{2} - 6 y + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.7.2

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

$2 x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.8

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 1.2.8.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$2 x + 4 \frac{d}{d x} [x] + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.8.2

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

$2 x + 4 \cdot 1 + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.8.3

Multiply $4$ by $1$ .

$2 x + 4 + \frac{d}{d x} [4] + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.9

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$2 x + 4 + 0 + \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.10

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

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Step 1.2.10.1

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

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Step 1.2.10.1.1

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

$2 x + 4 + 0 + \frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.10.1.2

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

$2 x + 4 + 0 + 2 u \frac{d}{d x} [y] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.10.1.3

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

$2 x + 4 + 0 + 2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.10.2

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

$2 x + 4 + 0 + 2 y y' + \frac{d}{d x} [- 6 y] + \frac{d}{d x} [9]$

Step 1.2.11

Evaluate $\frac{d}{d x} [- 6 y]$ .

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Step 1.2.11.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 y$ with respect to $x$ is $- 6 \frac{d}{d x} [y]$ .

$2 x + 4 + 0 + 2 y y' - 6 \frac{d}{d x} [y] + \frac{d}{d x} [9]$

Step 1.2.11.2

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

$2 x + 4 + 0 + 2 y y' - 6 y' + \frac{d}{d x} [9]$

Step 1.2.12

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 4 + 0 + 2 y y' - 6 y' + 0$

Step 1.2.13

Simplify.

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Step 1.2.13.1

Combine terms.

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Step 1.2.13.1.1

Add $2 x + 4$ and $0$ .

$2 x + 4 + 2 y y' - 6 y' + 0$

Step 1.2.13.1.2

Add $2 x + 4 + 2 y y' - 6 y'$ and $0$ .

$2 x + 4 + 2 y y' - 6 y'$

Step 1.2.13.2

Reorder terms.

$2 y y' + 2 x + 4 - 6 y'$

Step 1.3

Since $37$ is constant with respect to $x$ , the derivative of $37$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$2 y y' + 2 x + 4 - 6 y' = 0$

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.

$2 y y' + 4 - 6 y' = - 2 x$

Step 1.5.1.2

Subtract $4$ from both sides of the equation.

$2 y y' - 6 y' = - 2 x - 4$

Step 1.5.2

Factor $2 y'$ out of $2 y y' - 6 y'$ .

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Step 1.5.2.1

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

$2 y' y - 6 y' = - 2 x - 4$

Step 1.5.2.2

Factor $2 y'$ out of $- 6 y'$ .

$2 y' y + 2 y' \cdot - 3 = - 2 x - 4$

Step 1.5.2.3

Factor $2 y'$ out of $2 y' y + 2 y' \cdot - 3$ .

$2 y' (y - 3) = - 2 x - 4$

Step 1.5.3

Divide each term in $2 y' (y - 3) = - 2 x - 4$ by $2 (y - 3)$ and simplify.

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Step 1.5.3.1

Divide each term in $2 y' (y - 3) = - 2 x - 4$ by $2 (y - 3)$ .

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

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.

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

Step 1.5.3.2.1.2

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of $y - 3$ .

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Step 1.5.3.2.2.1

Cancel the common factor.

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

Step 1.5.3.2.2.2

Divide $y'$ by $1$ .

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

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 $- 2 x$ .

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

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.

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

Step 1.5.3.3.1.1.2.2

Rewrite the expression.

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

Step 1.5.3.3.1.2

Move the negative in front of the fraction.

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

Step 1.5.3.3.1.3

Cancel the common factor of $- 4$ and $2$ .

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Step 1.5.3.3.1.3.1

Factor $2$ out of $- 4$ .

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

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.

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

Step 1.5.3.3.1.3.2.2

Rewrite the expression.

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

Step 1.5.3.3.1.4

Move the negative in front of the fraction.

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

Step 1.5.3.3.2

Simplify terms.

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Step 1.5.3.3.2.1

Combine the numerators over the common denominator.

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

Step 1.5.3.3.2.2

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

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

Step 1.5.3.3.2.3

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

$y' = \frac{- (x) - 1 (2)}{y - 3}$

Step 1.5.3.3.2.4

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

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

Step 1.5.3.3.2.5

Simplify the expression.

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Step 1.5.3.3.2.5.1

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

$y' = \frac{- 1 (x + 2)}{y - 3}$

Step 1.5.3.3.2.5.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Add $4$ and $2$ .

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

Step 1.7.4

Subtract $3$ from $4$ .

$- \frac{6}{1}$

Step 1.7.5

Divide $6$ by $1$ .

$- 1 \cdot 6$

Step 1.7.6

Multiply $- 1$ by $6$ .

$- 6$

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

$y - (4) = - 6 \cdot (x - (4))$

Step 2.2

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

$y - 4 = - 6 \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 6 \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 4 = 0 + 0 - 6 \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 = - 6 \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 4 = - 6 x - 6 \cdot - 4$

Step 2.3.1.4

Multiply $- 6$ by $- 4$ .

$y - 4 = - 6 x + 24$

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.

$y = - 6 x + 24 + 4$

Step 2.3.2.2

Add $24$ and $4$ .

$y = - 6 x + 28$

Step 3