Calculus Examples

$f (x) = \frac{x - 3}{4 + x}$ , $[- 4, 1]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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 $f (x) = x - 3$ and $g (x) = 4 + x$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

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

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

Step 1.1.1.2.3

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

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

Step 1.1.1.2.4

Simplify the expression.

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Step 1.1.1.2.4.1

Add $1$ and $0$ .

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

Step 1.1.1.2.4.2

Multiply $4 + x$ by $1$ .

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

Step 1.1.1.2.5

By the Sum Rule, the derivative of $4 + x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [x]$ .

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

Step 1.1.1.2.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$\frac{4 + x - (x - 3) (0 + \frac{d}{d x} [x])}{{(4 + x)}^{2}}$

Step 1.1.1.2.7

Add $0$ and $\frac{d}{d x} [x]$ .

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

Step 1.1.1.2.8

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

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

Step 1.1.1.2.9

Multiply $- 1$ by $1$ .

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

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Simplify the numerator.

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Step 1.1.1.3.2.1

Combine the opposite terms in $4 + x - x - - 3$ .

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Step 1.1.1.3.2.1.1

Subtract $x$ from $x$ .

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

Step 1.1.1.3.2.1.2

Add $4$ and $0$ .

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

Step 1.1.1.3.2.2

Multiply $- 1$ by $- 3$ .

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

Step 1.1.1.3.2.3

Add $4$ and $3$ .

$\frac{7}{{(4 + x)}^{2}}$

Step 1.1.1.3.3

Reorder terms.

$f' (x) = \frac{7}{{(x + 4)}^{2}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{7}{{(x + 4)}^{2}}$ .

$\frac{7}{{(x + 4)}^{2}}$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $\frac{7}{{(x + 4)}^{2}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$\frac{7}{{(x + 4)}^{2}} = 0$

Step 1.2.2

Set the numerator equal to zero.

$7 = 0$

Step 1.2.3

Since $7 \neq 0$ , there are no solutions.

No solution

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{7}{{(x + 4)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x + 4)}^{2} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.3.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.4

Evaluate $\frac{x - 3}{4 + x}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = - 4$ .

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Step 1.4.1.1

Substitute $- 4$ for $x$ .

$\frac{(- 4) - 3}{4 - 4}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Remove parentheses.

$\frac{(- 4) - 3}{4 - 4}$

Step 1.4.1.2.2

Subtract $4$ from $4$ .

$\frac{(- 4) - 3}{0}$

Step 1.4.1.2.3

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

Undefined

Step 1.5

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

No critical points found

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 4$ .

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Step 2.1.1

Substitute $- 4$ for $x$ .

$\frac{(- 4) - 3}{4 - 4}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Remove parentheses.

$\frac{(- 4) - 3}{4 - 4}$

Step 2.1.2.2

Subtract $4$ from $4$ .

$\frac{(- 4) - 3}{0}$

Step 2.1.2.3

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

Undefined

Step 2.2

Evaluate at $x = 1$ .

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Step 2.2.1

Substitute $1$ for $x$ .

$\frac{(1) - 3}{4 + 1}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Remove parentheses.

$\frac{(1) - 3}{4 + 1}$

Step 2.2.2.2

Subtract $3$ from $1$ .

$\frac{- 2}{4 + 1}$

Step 2.2.2.3

Add $4$ and $1$ .

$\frac{- 2}{5}$

Step 2.2.2.4

Move the negative in front of the fraction.

$- \frac{2}{5}$

Step 2.3

List all of the points.

$(1, - \frac{2}{5})$

Step 3

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(1, - \frac{2}{5})$

No absolute minimum

Step 5