Calculus Examples

$y = \frac{4 x}{x - 3}$

Step 1

Differentiate both sides of the equation.

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

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

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

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 $f (x) = x$ and $g (x) = x - 3$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

$4 \frac{(x - 3) \cdot 1 - x \frac{d}{d x} [x - 3]}{{(x - 3)}^{2}}$

Step 3.3.2

Multiply $x - 3$ by $1$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

$4 \frac{x - 3 - x (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{2}}$

Step 3.3.5

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

$4 \frac{x - 3 - x (1 + 0)}{{(x - 3)}^{2}}$

Step 3.3.6

Simplify terms.

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Step 3.3.6.1

Add $1$ and $0$ .

$4 \frac{x - 3 - x \cdot 1}{{(x - 3)}^{2}}$

Step 3.3.6.2

Multiply $- 1$ by $1$ .

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

Step 3.3.6.3

Subtract $x$ from $x$ .

$4 \frac{0 - 3}{{(x - 3)}^{2}}$

Step 3.3.6.4

Simplify the expression.

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Step 3.3.6.4.1

Subtract $3$ from $0$ .

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

Step 3.3.6.4.2

Move the negative in front of the fraction.

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

Step 3.3.6.4.3

Multiply $- 1$ by $4$ .

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

Step 3.3.6.5

Combine $- 4$ and $\frac{3}{{(x - 3)}^{2}}$ .

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

Step 3.3.6.6

Simplify the expression.

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Step 3.3.6.6.1

Multiply $- 4$ by $3$ .

$\frac{- 12}{{(x - 3)}^{2}}$

Step 3.3.6.6.2

Move the negative in front of the fraction.

$- \frac{12}{{(x - 3)}^{2}}$

Step 4

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

$y' = - \frac{12}{{(x - 3)}^{2}}$

Step 5

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

$\frac{d y}{d x} = - \frac{12}{{(x - 3)}^{2}}$