Calculus Examples

${(y - 3)}^{2} = 4 (x - 5)$ , $(6, 5)$

Step 1

Find the first derivative and evaluate at $x = 6$ and $y = 5$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(y - 3)}^{2}) = \frac{d}{d x} (4 (x - 5))$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Apply the distributive property.

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

Step 1.2.2.3

Apply the distributive property.

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

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 $y$ by $y$ .

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

Step 1.2.3.1.2

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

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

Step 1.2.3.1.3

Multiply $- 3$ by $- 3$ .

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

Step 1.2.3.2

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

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

Step 1.2.4

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

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

Step 1.2.5

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 1.2.5.1

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

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

Step 1.2.5.2

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

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

Step 1.2.5.3

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

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

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

Step 1.2.9

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

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

Step 1.2.10

Add $2 y y' - 6 y'$ and $0$ .

$2 y y' - 6 y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 (x - 5)$ with respect to $x$ is $4 \frac{d}{d x} [x - 5]$ .

$4 \frac{d}{d x} [x - 5]$

Step 1.3.2

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$4 (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 1.3.3

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

$4 (1 + \frac{d}{d x} [- 5])$

Step 1.3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$4 (1 + 0)$

Step 1.3.5

Simplify the expression.

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Step 1.3.5.1

Add $1$ and $0$ .

$4 \cdot 1$

Step 1.3.5.2

Multiply $4$ by $1$ .

$4$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

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

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Step 1.5.1.1

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

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

Step 1.5.1.2

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

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

Step 1.5.1.3

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

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.

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

Step 1.5.2.2.1.2

Rewrite the expression.

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

Step 1.5.2.2.2

Cancel the common factor of $y - 3$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

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

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Step 1.5.2.3.1.1

Factor $2$ out of $4$ .

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

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

Cancel the common factor.

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

Step 1.5.2.3.1.2.2

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 6$ and $y = 5$ .

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Step 1.7.1

Replace the variable $x$ with $6$ in the expression.

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

Step 1.7.2

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

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

Step 1.7.3

Subtract $3$ from $5$ .

$\frac{2}{2}$

Step 1.7.4

Divide $2$ by $2$ .

$1$

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 $1$ and a given point $(6, 5)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (5) = 1 \cdot (x - (6))$

Step 2.2

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

$y - 5 = 1 \cdot (x - 6)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x - 6$ by $1$ .

$y - 5 = x - 6$

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 $5$ to both sides of the equation.

$y = x - 6 + 5$

Step 2.3.2.2

Add $- 6$ and $5$ .

$y = x - 1$

Step 3