Calculus Examples

$y = \frac{x - 6}{4 x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{x - 6}{4 x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x - 6}{4 x}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{x - 6}{x}]$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x - 6}{x}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 6$ and $g (x) = x$ .

$\frac{1}{4} \cdot \frac{x \frac{d}{d x} [x - 6] - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{1}{4} \cdot \frac{x (1 + 0) - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.4

Simplify the expression.

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Step 3.3.4.1

Add $1$ and $0$ .

$\frac{1}{4} \cdot \frac{x \cdot 1 - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.4.2

Multiply $x$ by $1$ .

$\frac{1}{4} \cdot \frac{x - (x - 6) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.5

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

$\frac{1}{4} \cdot \frac{x - (x - 6) \cdot 1}{x^{2}}$

Step 3.3.6

Combine fractions.

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Step 3.3.6.1

Multiply $- 1$ by $1$ .

$\frac{1}{4} \cdot \frac{x - (x - 6)}{x^{2}}$

Step 3.3.6.2

Multiply $\frac{1}{4}$ by $\frac{x - (x - 6)}{x^{2}}$ .

$\frac{x - (x - 6)}{4 x^{2}}$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$\frac{x - x - - 6}{4 x^{2}}$

Step 3.4.2

Simplify the numerator.

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Step 3.4.2.1

Subtract $x$ from $x$ .

$\frac{0 - - 6}{4 x^{2}}$

Step 3.4.2.2

Subtract $- 6$ from $0$ .

$\frac{- - 6}{4 x^{2}}$

Step 3.4.2.3

Multiply $- 1$ by $- 6$ .

$\frac{6}{4 x^{2}}$

Step 3.4.3

Cancel the common factor of $6$ and $4$ .

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Step 3.4.3.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{4 x^{2}}$

Step 3.4.3.2

Cancel the common factors.

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Step 3.4.3.2.1

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

$\frac{2 (3)}{2 (2 x^{2})}$

Step 3.4.3.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 (2 x^{2})}$

Step 3.4.3.2.3

Rewrite the expression.

$\frac{3}{2 x^{2}}$

Step 4

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

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

Step 5

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

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