Calculus Examples

$f (x) = \frac{- 4 x}{x + 9}$

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

Differentiate using the Constant Multiple Rule.

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Step 1.1.1.1

Move the negative in front of the fraction.

$\frac{d}{d x} [- \frac{4 x}{x + 9}]$

Step 1.1.1.2

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4 x}{x + 9}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{x}{x + 9}]$ .

$- 4 \frac{d}{d x} [\frac{x}{x + 9}]$

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 $g (x) = x + 9$ .

$- 4 \frac{(x + 9) \frac{d}{d x} [x] - x \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

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 $n = 1$ .

$- 4 \frac{(x + 9) \cdot 1 - x \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.1.3.2

Multiply $x + 9$ by $1$ .

$- 4 \frac{x + 9 - x \frac{d}{d x} [x + 9]}{{(x + 9)}^{2}}$

Step 1.1.3.3

By the Sum Rule, the derivative of $x + 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [9]$ .

$- 4 \frac{x + 9 - x (\frac{d}{d x} [x] + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

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 $n = 1$ .

$- 4 \frac{x + 9 - x (1 + \frac{d}{d x} [9])}{{(x + 9)}^{2}}$

Step 1.1.3.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$- 4 \frac{x + 9 - x (1 + 0)}{{(x + 9)}^{2}}$

Step 1.1.3.6

Simplify terms.

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Step 1.1.3.6.1

Add $1$ and $0$ .

$- 4 \frac{x + 9 - x \cdot 1}{{(x + 9)}^{2}}$

Step 1.1.3.6.2

Multiply $- 1$ by $1$ .

$- 4 \frac{x + 9 - x}{{(x + 9)}^{2}}$

Step 1.1.3.6.3

Subtract $x$ from $x$ .

$- 4 \frac{0 + 9}{{(x + 9)}^{2}}$

Step 1.1.3.6.4

Add $0$ and $9$ .

$- 4 \frac{9}{{(x + 9)}^{2}}$

Step 1.1.3.6.5

Combine $- 4$ and $\frac{9}{{(x + 9)}^{2}}$ .

$\frac{- 4 \cdot 9}{{(x + 9)}^{2}}$

Step 1.1.3.6.6

Simplify the expression.

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Step 1.1.3.6.6.1

Multiply $- 4$ by $9$ .

$\frac{- 36}{{(x + 9)}^{2}}$

Step 1.1.3.6.6.2

Move the negative in front of the fraction.

$f' (x) = - \frac{36}{{(x + 9)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{36}{{(x + 9)}^{2}}$ .

$- \frac{36}{{(x + 9)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{36}{{(x + 9)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{36}{{(x + 9)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$36 = 0$

Step 2.3

Since $36 \neq 0$ , 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 $\frac{36}{{(x + 9)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 9)}^{2} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 3.2.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 4

Evaluate $\frac{- 4 x}{x + 9}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - 9$ .

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Step 4.1.1

Substitute $- 9$ for $x$ .

$\frac{- 4 \cdot - 9}{(- 9) + 9}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Add $- 9$ and $9$ .

$\frac{- 4 \cdot - 9}{0}$

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