Calculus Examples

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

Step 1

Solve the equation as $y$ in terms of $x$ .

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Step 1.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y - 2 = \pm \sqrt{4 (x - 3)}$

Step 1.2

Simplify $\pm \sqrt{4 (x - 3)}$ .

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Step 1.2.1

Rewrite $4$ as $2^{2}$ .

$y - 2 = \pm \sqrt{2^{2} (x - 3)}$

Step 1.2.2

Pull terms out from under the radical.

$y - 2 = \pm 2 \sqrt{x - 3}$

Step 1.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.3.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 2 = 2 \sqrt{x - 3}$

Step 1.3.2

Add $2$ to both sides of the equation.

$y = 2 \sqrt{x - 3} + 2$

Step 1.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$y - 2 = - 2 \sqrt{x - 3}$

Step 1.3.4

Add $2$ to both sides of the equation.

$y = - 2 \sqrt{x - 3} + 2$

Step 1.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = 2 \sqrt{x - 3} + 2$

$y = - 2 \sqrt{x - 3} + 2$

$y = 2 \sqrt{x - 3} + 2$

$y = - 2 \sqrt{x - 3} + 2$

$y = 2 \sqrt{x - 3} + 2$

$y = - 2 \sqrt{x - 3} + 2$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = 2 \sqrt{x - 3} + 2 \to f (x) = 2 \sqrt{x - 3} + 2$

$y = - 2 \sqrt{x - 3} + 2 \to f (x) = - 2 \sqrt{x - 3} + 2$

Step 3

Because the $y$ variable in the equation ${(y - 2)}^{2} = 4 (x - 3)$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

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Step 3.1

Differentiate both sides of the equation.

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

Step 3.2

Differentiate the left side of the equation.

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Step 3.2.1

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

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

Step 3.2.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

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

Step 3.2.2.2

Apply the distributive property.

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

Step 3.2.2.3

Apply the distributive property.

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

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply $y$ by $y$ .

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

Step 3.2.3.1.2

Move $- 2$ to the left of $y$ .

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

Step 3.2.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.2.3.2

Subtract $2 y$ from $- 2 y$ .

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

Step 3.2.4

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

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

Step 3.2.5

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 3.2.5.1

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

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

Step 3.2.5.2

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

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

Step 3.2.5.3

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

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

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

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

Step 3.2.9

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

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

Step 3.2.10

Add $2 y y' - 4 y'$ and $0$ .

$2 y y' - 4 y'$

Step 3.3

Differentiate the right side of the equation.

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Step 3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 (x - 3)$ with respect to $x$ is $4 \frac{d}{d x} [x - 3]$ .

$4 \frac{d}{d x} [x - 3]$

Step 3.3.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

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

Step 3.3.3

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

$4 (1 + \frac{d}{d x} [- 3])$

Step 3.3.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$4 (1 + 0)$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Add $1$ and $0$ .

$4 \cdot 1$

Step 3.3.5.2

Multiply $4$ by $1$ .

$4$

Step 3.4

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

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

Step 3.5

Solve for $y'$ .

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Step 3.5.1

Factor $2 y'$ out of $2 y y' - 4 y'$ .

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Step 3.5.1.1

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

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

Step 3.5.1.2

Factor $2 y'$ out of $- 4 y'$ .

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

Step 3.5.1.3

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

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.1.2

Rewrite the expression.

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

Step 3.5.2.2.2

Cancel the common factor of $y - 2$ .

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Step 3.5.2.2.2.1

Cancel the common factor.

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

Step 3.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

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

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Step 3.5.2.3.1.1

Factor $2$ out of $4$ .

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

Step 3.5.2.3.1.2

Cancel the common factors.

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Step 3.5.2.3.1.2.1

Cancel the common factor.

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

Step 3.5.2.3.1.2.2

Rewrite the expression.

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

Step 3.6

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

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

Step 4

Set the derivative equal to $0$ then solve the equation $\frac{2}{y - 2} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$2 = 0$

Step 4.2

Since $2 \neq 0$ , there are no solutions.

No solution

Step 5

There are no solution found by setting the derivative equal to $0$ , $\frac{2}{y - 2} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 6