Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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 $g (x) = x - 6$ .

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

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

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

Step 1.1.3.2

Multiply $x - 6$ by $1$ .

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

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

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

Step 1.1.3.6

Simplify terms.

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Step 1.1.3.6.1

Add $1$ and $0$ .

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

Step 1.1.3.6.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.6.3

Subtract $x$ from $x$ .

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

Step 1.1.3.6.4

Simplify the expression.

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Step 1.1.3.6.4.1

Subtract $6$ from $0$ .

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

Step 1.1.3.6.4.2

Move the negative in front of the fraction.

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

Step 1.1.3.6.4.3

Multiply $- 1$ by $4$ .

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

Step 1.1.3.6.5

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

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

Step 1.1.3.6.6

Simplify the expression.

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Step 1.1.3.6.6.1

Multiply $- 4$ by $6$ .

$\frac{- 24}{{(x - 6)}^{2}}$

Step 1.1.3.6.6.2

Move the negative in front of the fraction.

$f' (x) = - \frac{24}{{(x - 6)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{24}{{(x - 6)}^{2}}$ .

$- \frac{24}{{(x - 6)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{24}{{(x - 6)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{24}{{(x - 6)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$24 = 0$

Step 2.3

Since $24 \neq 0$ , there are no solutions.

No solution

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{24}{{(x - 6)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 6)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Evaluate $\frac{4 x}{x - 6}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 6$ .

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Step 4.1.1

Substitute $6$ for $x$ .

$\frac{4 (6)}{(6) - 6}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $6$ from $6$ .

$\frac{4 (6)}{0}$

Step 4.1.2.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

Step 5

There are no values of $x$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found