Calculus Examples

$y = \frac{4 x^{2} - 16}{x - 2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{4 x^{2} - 16}{x - 2})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Factor $4$ out of $4 x^{2} - 16$ .

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Step 3.1.1.1

Factor $4$ out of $4 x^{2}$ .

$\frac{d}{d y} [\frac{4 (x^{2}) - 16}{x - 2}]$

Step 3.1.1.2

Factor $4$ out of $- 16$ .

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

Step 3.1.1.3

Factor $4$ out of $4 x^{2} + 4 \cdot - 4$ .

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

Step 3.1.2

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

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

Step 3.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $x - 2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $4 (x + 2)$ by $1$ .

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Multiply $4$ by $2$ .

$\frac{d}{d y} [4 x + 8]$

Step 3.3

By the Sum Rule, the derivative of $4 x + 8$ with respect to $y$ is $\frac{d}{d y} [4 x] + \frac{d}{d y} [8]$ .

$\frac{d}{d y} [4 x] + \frac{d}{d y} [8]$

Step 3.4

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

$4 \frac{d}{d y} [x] + \frac{d}{d y} [8]$

Step 3.5

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

$4 x' + \frac{d}{d y} [8]$

Step 3.6

Since $8$ is constant with respect to $y$ , the derivative of $8$ with respect to $y$ is $0$ .

$4 x' + 0$

Step 3.7

Add $4 x'$ and $0$ .

$4 x'$

Step 4

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

$1 = 4 x'$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $4 x' = 1$ .

$4 x' = 1$

Step 5.2

Divide each term in $4 x' = 1$ by $4$ and simplify.

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Step 5.2.1

Divide each term in $4 x' = 1$ by $4$ .

$\frac{4 x'}{4} = \frac{1}{4}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $4$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{4 x'}{4} = \frac{1}{4}$

Step 5.2.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{1}{4}$

Step 6

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

$\frac{d x}{d y} = \frac{1}{4}$