Calculus Examples

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

Step 1

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

$f (x) = \frac{x}{x - 3}$

Step 2

Find the derivative.

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Step 2.1

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 $g (x) = x - 3$ .

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

Multiply $x - 3$ by $1$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify by adding terms.

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Step 2.2.6.1

Add $1$ and $0$ .

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

Step 2.2.6.2

Multiply $- 1$ by $1$ .

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

Step 2.2.6.3

Subtract $x$ from $x$ .

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

Step 2.2.6.4

Simplify the expression.

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Step 2.2.6.4.1

Subtract $3$ from $0$ .

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

Step 2.2.6.4.2

Move the negative in front of the fraction.

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

Step 3

Set the derivative equal to $0$ then solve the equation $- \frac{3}{{(x - 3)}^{2}} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$3 = 0$

Step 3.2

Since $3 \neq 0$ , there are no solutions.

No solution

Step 4

There are no solution found by setting the derivative equal to $0$ , $- \frac{3}{{(x - 3)}^{2}} = 0$ so there are no horizontal tangent lines.

No horizontal tangent lines found

Step 5