Calculus Examples

$\frac{x^{2}}{{(x - 7)}^{2}}$

Step 1

Write $\frac{x^{2}}{{(x - 7)}^{2}}$ as a function.

$f (x) = \frac{x^{2}}{{(x - 7)}^{2}}$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

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 $f (x) = x^{2}$ and $g (x) = {(x - 7)}^{2}$ .

$f' (x) = \frac{{(x - 7)}^{2} \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} {(x - 7)}^{2}}{{({(x - 7)}^{2})}^{2}}$

Step 2.1.2

Differentiate using the Power Rule.

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Step 2.1.2.1

Multiply the exponents in ${({(x - 7)}^{2})}^{2}$ .

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Step 2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f' (x) = \frac{{(x - 7)}^{2} \frac{d}{d x} (x^{2}) - x^{2} \frac{d}{d x} {(x - 7)}^{2}}{{(x - 7)}^{2 \cdot 2}}$

Step 2.1.2.1.2

Multiply $2$ by $2$ .

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

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.1.2.3

Move $2$ to the left of ${(x - 7)}^{2}$ .

$f' (x) = \frac{2 \cdot ({(x - 7)}^{2} x) - x^{2} \frac{d}{d x} {(x - 7)}^{2}}{{(x - 7)}^{4}}$

Step 2.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x - 7$ .

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Step 2.1.3.1

To apply the Chain Rule, set $u$ as $x - 7$ .

$f' (x) = \frac{2 {(x - 7)}^{2} x - x^{2} (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x - 7))}{{(x - 7)}^{4}}$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{2 {(x - 7)}^{2} x - x^{2} (2 u \frac{d}{d x} (x - 7))}{{(x - 7)}^{4}}$

Step 2.1.3.3

Replace all occurrences of $u$ with $x - 7$ .

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

Step 2.1.4

Differentiate.

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Step 2.1.4.1

Multiply $2$ by $- 1$ .

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

Step 2.1.4.2

By the Sum Rule, the derivative of $x - 7$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 7]$ .

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

Step 2.1.4.3

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

$f' (x) = \frac{2 {(x - 7)}^{2} x - 2 x^{2} ((x - 7) (1 + \frac{d}{d x} (- 7)))}{{(x - 7)}^{4}}$

Step 2.1.4.4

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f' (x) = \frac{2 {(x - 7)}^{2} x - 2 x^{2} ((x - 7) (1 + 0))}{{(x - 7)}^{4}}$

Step 2.1.4.5

Simplify the expression.

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Step 2.1.4.5.1

Add $1$ and $0$ .

$f' (x) = \frac{2 {(x - 7)}^{2} x - 2 x^{2} ((x - 7) \cdot 1)}{{(x - 7)}^{4}}$

Step 2.1.4.5.2

Multiply $x - 7$ by $1$ .

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

Step 2.1.5

Simplify.

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Step 2.1.5.1

Apply the distributive property.

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

Step 2.1.5.2

Simplify the numerator.

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Step 2.1.5.2.1

Simplify each term.

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Step 2.1.5.2.1.1

Rewrite ${(x - 7)}^{2}$ as $(x - 7) (x - 7)$ .

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

Step 2.1.5.2.1.2

Expand $(x - 7) (x - 7)$ using the FOIL Method.

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Step 2.1.5.2.1.2.1

Apply the distributive property.

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

Step 2.1.5.2.1.2.2

Apply the distributive property.

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

Step 2.1.5.2.1.2.3

Apply the distributive property.

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

Step 2.1.5.2.1.3

Simplify and combine like terms.

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Step 2.1.5.2.1.3.1

Simplify each term.

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Step 2.1.5.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.5.2.1.3.1.2

Move $- 7$ to the left of $x$ .

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

Step 2.1.5.2.1.3.1.3

Multiply $- 7$ by $- 7$ .

$f' (x) = \frac{2 (x^{2} - 7 x - 7 x + 49) x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.3.2

Subtract $7 x$ from $- 7 x$ .

$f' (x) = \frac{2 (x^{2} - 14 x + 49) x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.4

Apply the distributive property.

$f' (x) = \frac{(2 x^{2} + 2 (- 14 x) + 2 \cdot 49) x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.5

Simplify.

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Step 2.1.5.2.1.5.1

Multiply $- 14$ by $2$ .

$f' (x) = \frac{(2 x^{2} - 28 x + 2 \cdot 49) x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.5.2

Multiply $2$ by $49$ .

$f' (x) = \frac{(2 x^{2} - 28 x + 98) x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.6

Apply the distributive property.

$f' (x) = \frac{2 x^{2} x - 28 x \cdot x + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7

Simplify.

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Step 2.1.5.2.1.7.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.5.2.1.7.1.1

Move $x$ .

$f' (x) = \frac{2 (x \cdot x^{2}) - 28 x \cdot x + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.5.2.1.7.1.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 (x \cdot x^{2}) - 28 x \cdot x + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{2 x^{1 + 2} - 28 x \cdot x + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7.1.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} - 28 x \cdot x + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.5.2.1.7.2.1

Move $x$ .

$f' (x) = \frac{2 x^{3} - 28 (x \cdot x) + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.7.2.2

Multiply $x$ by $x$ .

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 x^{2} x - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.8

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.5.2.1.8.1

Move $x$ .

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 (x \cdot x^{2}) - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.8.2

Multiply $x$ by $x^{2}$ .

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Step 2.1.5.2.1.8.2.1

Raise $x$ to the power of $1$ .

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 (x \cdot x^{2}) - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 x^{1 + 2} - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.8.3

Add $1$ and $2$ .

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 x^{3} - 2 x^{2} \cdot - 7}{{(x - 7)}^{4}}$

Step 2.1.5.2.1.9

Multiply $- 7$ by $- 2$ .

$f' (x) = \frac{2 x^{3} - 28 x^{2} + 98 x - 2 x^{3} + 14 x^{2}}{{(x - 7)}^{4}}$

Step 2.1.5.2.2

Combine the opposite terms in $2 x^{3} - 28 x^{2} + 98 x - 2 x^{3} + 14 x^{2}$ .

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Step 2.1.5.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$f' (x) = \frac{- 28 x^{2} + 98 x + 0 + 14 x^{2}}{{(x - 7)}^{4}}$

Step 2.1.5.2.2.2

Add $- 28 x^{2} + 98 x$ and $0$ .

$f' (x) = \frac{- 28 x^{2} + 98 x + 14 x^{2}}{{(x - 7)}^{4}}$

Step 2.1.5.2.3

Add $- 28 x^{2}$ and $14 x^{2}$ .

$f' (x) = \frac{- 14 x^{2} + 98 x}{{(x - 7)}^{4}}$

Step 2.1.5.3

Factor $14 x$ out of $- 14 x^{2} + 98 x$ .

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Step 2.1.5.3.1

Factor $14 x$ out of $- 14 x^{2}$ .

$f' (x) = \frac{14 x (- x) + 98 x}{{(x - 7)}^{4}}$

Step 2.1.5.3.2

Factor $14 x$ out of $98 x$ .

$f' (x) = \frac{14 x (- x) + 14 x (7)}{{(x - 7)}^{4}}$

Step 2.1.5.3.3

Factor $14 x$ out of $14 x (- x) + 14 x (7)$ .

$f' (x) = \frac{14 x (- x + 7)}{{(x - 7)}^{4}}$

Step 2.1.5.4

Cancel the common factor of $- x + 7$ and ${(x - 7)}^{4}$ .

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Step 2.1.5.4.1

Factor $- 1$ out of $- x$ .

$f' (x) = \frac{14 x (- (x) + 7)}{{(x - 7)}^{4}}$

Step 2.1.5.4.2

Rewrite $7$ as $- 1 (- 7)$ .

$f' (x) = \frac{14 x (- (x) - 1 \cdot - 7)}{{(x - 7)}^{4}}$

Step 2.1.5.4.3

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$f' (x) = \frac{14 x (- (x - 7))}{{(x - 7)}^{4}}$

Step 2.1.5.4.4

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$f' (x) = \frac{14 x (- 1 (x - 7))}{{(x - 7)}^{4}}$

Step 2.1.5.4.5

Factor $x - 7$ out of $14 x (- 1 (x - 7))$ .

$f' (x) = \frac{(x - 7) (14 x \cdot (- 1))}{{(x - 7)}^{4}}$

Step 2.1.5.4.6

Cancel the common factors.

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Step 2.1.5.4.6.1

Factor $x - 7$ out of ${(x - 7)}^{4}$ .

$f' (x) = \frac{(x - 7) (14 x \cdot (- 1))}{(x - 7) {(x - 7)}^{3}}$

Step 2.1.5.4.6.2

Cancel the common factor.

$f' (x) = \frac{(x - 7) (14 x \cdot (- 1))}{(x - 7) {(x - 7)}^{3}}$

Step 2.1.5.4.6.3

Rewrite the expression.

$f' (x) = \frac{14 x \cdot (- 1)}{{(x - 7)}^{3}}$

Step 2.1.5.5

Multiply $- 1$ by $14$ .

$f' (x) = \frac{- 14 x}{{(x - 7)}^{3}}$

Step 2.1.5.6

Move the negative in front of the fraction.

$f' (x) = - \frac{14 x}{{(x - 7)}^{3}}$

Step 2.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{14 x}{{(x - 7)}^{3}}$ .

$- \frac{14 x}{{(x - 7)}^{3}}$

Step 3

Set the first derivative equal to $0$ then solve the equation $- \frac{14 x}{{(x - 7)}^{3}} = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$- \frac{14 x}{{(x - 7)}^{3}} = 0$

Step 3.2

Set the numerator equal to zero.

$14 x = 0$

Step 3.3

Divide each term in $14 x = 0$ by $14$ and simplify.

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Step 3.3.1

Divide each term in $14 x = 0$ by $14$ .

$\frac{14 x}{14} = \frac{0}{14}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $14$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{14 x}{14} = \frac{0}{14}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{14}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $14$ .

$x = 0$

Step 4

The values which make the derivative equal to $0$ are $0$ .

$0$

Step 5

Find where the derivative is undefined.

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Step 5.1

Set the denominator in $\frac{14 x}{{(x - 7)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x - 7)}^{3} = 0$

Step 5.2

Solve for $x$ .

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Step 5.2.1

Set the $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 5.2.2

Add $7$ to both sides of the equation.

$x = 7$

Step 6

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 7) \cup (7, \infty)$

Step 7

Substitute a value from the interval $(- \infty, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = - \frac{14 (- 1)}{{((- 1) - 7)}^{3}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $14$ by $- 1$ .

$f' (- 1) = - \frac{- 14}{{(- 1 - 7)}^{3}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Subtract $7$ from $- 1$ .

$f' (- 1) = - \frac{- 14}{{(- 8)}^{3}}$

Step 7.2.2.2

Raise $- 8$ to the power of $3$ .

$f' (- 1) = - \frac{- 14}{- 512}$

Step 7.2.3

Cancel the common factor of $- 14$ and $- 512$ .

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Step 7.2.3.1

Factor $- 2$ out of $- 14$ .

$f' (- 1) = - \frac{- 2 \cdot 7}{- 512}$

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $- 2$ out of $- 512$ .

$f' (- 1) = - \frac{- 2 \cdot 7}{- 2 \cdot 256}$

Step 7.2.3.2.2

Cancel the common factor.

$f' (- 1) = - \frac{- 2 \cdot 7}{- 2 \cdot 256}$

Step 7.2.3.2.3

Rewrite the expression.

$f' (- 1) = - \frac{7}{256}$

Step 7.2.4

The final answer is $- \frac{7}{256}$ .

$- \frac{7}{256}$

Step 7.3

At $x = - 1$ the derivative is $- \frac{7}{256}$ . Since this is negative, the function is decreasing on $(- \infty, 0)$ .

Decreasing on $(- \infty, 0)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(0, 7)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f' (\frac{7}{2}) = - \frac{14 (\frac{7}{2})}{{((\frac{7}{2}) - 7)}^{3}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Combine $14$ and $\frac{7}{2}$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{7}{2} - 7)}^{3}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{7}{2} - 7 \cdot \frac{2}{2})}^{3}}$

Step 8.2.2.2

Combine $- 7$ and $\frac{2}{2}$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{7}{2} + \frac{- 7 \cdot 2}{2})}^{3}}$

Step 8.2.2.3

Combine the numerators over the common denominator.

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{7 - 7 \cdot 2}{2})}^{3}}$

Step 8.2.2.4

Simplify the numerator.

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Step 8.2.2.4.1

Multiply $- 7$ by $2$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{7 - 14}{2})}^{3}}$

Step 8.2.2.4.2

Subtract $14$ from $7$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(\frac{- 7}{2})}^{3}}$

Step 8.2.2.5

Move the negative in front of the fraction.

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(- \frac{7}{2})}^{3}}$

Step 8.2.2.6

Apply the product rule to $- \frac{7}{2}$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{{(- 1)}^{3} {(\frac{7}{2})}^{3}}$

Step 8.2.2.7

Raise $- 1$ to the power of $3$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{- {(\frac{7}{2})}^{3}}$

Step 8.2.2.8

Apply the product rule to $\frac{7}{2}$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{- \frac{7^{3}}{2^{3}}}$

Step 8.2.2.9

Raise $7$ to the power of $3$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{- \frac{343}{2^{3}}}$

Step 8.2.2.10

Raise $2$ to the power of $3$ .

$f' (\frac{7}{2}) = - \frac{\frac{14 \cdot 7}{2}}{- \frac{343}{8}}$

Step 8.2.3

Simplify the expression.

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Step 8.2.3.1

Multiply $14$ by $7$ .

$f' (\frac{7}{2}) = - \frac{\frac{98}{2}}{- \frac{343}{8}}$

Step 8.2.3.2

Divide $98$ by $2$ .

$f' (\frac{7}{2}) = - \frac{49}{- \frac{343}{8}}$

Step 8.2.4

Multiply the numerator by the reciprocal of the denominator.

$f' (\frac{7}{2}) = - (49 (- \frac{8}{343}))$

Step 8.2.5

Cancel the common factor of $49$ .

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Step 8.2.5.1

Move the leading negative in $- \frac{8}{343}$ into the numerator.

$f' (\frac{7}{2}) = - (49 (\frac{- 8}{343}))$

Step 8.2.5.2

Factor $49$ out of $343$ .

$f' (\frac{7}{2}) = - (49 (\frac{- 8}{49 (7)}))$

Step 8.2.5.3

Cancel the common factor.

$f' (\frac{7}{2}) = - (49 (\frac{- 8}{49 \cdot 7}))$

Step 8.2.5.4

Rewrite the expression.

$f' (\frac{7}{2}) = - \frac{- 8}{7}$

Step 8.2.6

Move the negative in front of the fraction.

$f' (\frac{7}{2}) = \frac{8}{7}$

Step 8.2.7

The final answer is $\frac{8}{7}$ .

$\frac{8}{7}$

Step 8.3

At $x = \frac{7}{2}$ the derivative is $\frac{8}{7}$ . Since this is positive, the function is increasing on $(0, 7)$ .

Increasing on $(0, 7)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(7, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

Replace the variable $x$ with $8$ in the expression.

$f' (8) = - \frac{14 (8)}{{((8) - 7)}^{3}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $14$ by $8$ .

$f' (8) = - \frac{112}{{(8 - 7)}^{3}}$

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Subtract $7$ from $8$ .

$f' (8) = - \frac{112}{1^{3}}$

Step 9.2.2.2

One to any power is one.

$f' (8) = - \frac{112}{1}$

Step 9.2.3

Simplify the expression.

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Step 9.2.3.1

Divide $112$ by $1$ .

$f' (8) = - 1 \cdot 112$

Step 9.2.3.2

Multiply $- 1$ by $112$ .

$f' (8) = - 112$

Step 9.2.4

The final answer is $- 112$ .

$- 112$

Step 9.3

At $x = 8$ the derivative is $- 112$ . Since this is negative, the function is decreasing on $(7, \infty)$ .

Decreasing on $(7, \infty)$ since $f' (x) < 0$

Step 10

List the intervals on which the function is increasing and decreasing.

Increasing on: $(0, 7)$

Decreasing on: $(- \infty, 0), (7, \infty)$

Step 11