Calculus Examples

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

Step 1

Write $\frac{x^{2}}{{(x - 2)}^{3}}$ as a function.

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

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 $g (x) = {(x - 2)}^{3}$ .

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

Step 2.1.2

Differentiate using the Power Rule.

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Step 2.1.2.1

Multiply the exponents in ${({(x - 2)}^{3})}^{2}$ .

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Step 2.1.2.1.1

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

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

Step 2.1.2.1.2

Multiply $3$ by $2$ .

$\frac{{(x - 2)}^{3} \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

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 $n = 2$ .

$\frac{{(x - 2)}^{3} (2 x) - x^{2} \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

Step 2.1.2.3

Move $2$ to the left of ${(x - 2)}^{3}$ .

$\frac{2 \cdot {(x - 2)}^{3} x - x^{2} \frac{d}{d x} [{(x - 2)}^{3}]}{{(x - 2)}^{6}}$

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 $f (x) = x^{3}$ and $g (x) = x - 2$ .

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Step 2.1.3.1

To apply the Chain Rule, set $u$ as $x - 2$ .

$\frac{2 {(x - 2)}^{3} x - x^{2} (\frac{d}{d u} [u^{3}] \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

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 $n = 3$ .

$\frac{2 {(x - 2)}^{3} x - x^{2} (3 u^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 2.1.3.3

Replace all occurrences of $u$ with $x - 2$ .

$\frac{2 {(x - 2)}^{3} x - x^{2} (3 {(x - 2)}^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 2.1.4

Differentiate.

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Step 2.1.4.1

Multiply $3$ by $- 1$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} ({(x - 2)}^{2} \frac{d}{d x} [x - 2])}{{(x - 2)}^{6}}$

Step 2.1.4.2

By the Sum Rule, the derivative of $x - 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 2]$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} ({(x - 2)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]))}{{(x - 2)}^{6}}$

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 $n = 1$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} ({(x - 2)}^{2} (1 + \frac{d}{d x} [- 2]))}{{(x - 2)}^{6}}$

Step 2.1.4.4

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} ({(x - 2)}^{2} (1 + 0))}{{(x - 2)}^{6}}$

Step 2.1.4.5

Simplify the expression.

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Step 2.1.4.5.1

Add $1$ and $0$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} ({(x - 2)}^{2} \cdot 1)}{{(x - 2)}^{6}}$

Step 2.1.4.5.2

Multiply ${(x - 2)}^{2}$ by $1$ .

$\frac{2 {(x - 2)}^{3} x - 3 x^{2} {(x - 2)}^{2}}{{(x - 2)}^{6}}$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Simplify the numerator.

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Step 2.1.5.1.1

Factor ${(x - 2)}^{2} x$ out of $2 {(x - 2)}^{3} x - 3 x^{2} {(x - 2)}^{2}$ .

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Step 2.1.5.1.1.1

Factor ${(x - 2)}^{2} x$ out of $2 {(x - 2)}^{3} x$ .

$\frac{{(x - 2)}^{2} x (2 (x - 2)) - 3 x^{2} {(x - 2)}^{2}}{{(x - 2)}^{6}}$

Step 2.1.5.1.1.2

Factor ${(x - 2)}^{2} x$ out of $- 3 x^{2} {(x - 2)}^{2}$ .

$\frac{{(x - 2)}^{2} x (2 (x - 2)) + {(x - 2)}^{2} x (- 3 x)}{{(x - 2)}^{6}}$

Step 2.1.5.1.1.3

Factor ${(x - 2)}^{2} x$ out of ${(x - 2)}^{2} x (2 (x - 2)) + {(x - 2)}^{2} x (- 3 x)$ .

$\frac{{(x - 2)}^{2} x (2 (x - 2) - 3 x)}{{(x - 2)}^{6}}$

Step 2.1.5.1.2

Apply the distributive property.

$\frac{{(x - 2)}^{2} x (2 x + 2 \cdot - 2 - 3 x)}{{(x - 2)}^{6}}$

Step 2.1.5.1.3

Multiply $2$ by $- 2$ .

$\frac{{(x - 2)}^{2} x (2 x - 4 - 3 x)}{{(x - 2)}^{6}}$

Step 2.1.5.1.4

Subtract $3 x$ from $2 x$ .

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

Step 2.1.5.2

Cancel the common factor of ${(x - 2)}^{2}$ and ${(x - 2)}^{6}$ .

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Step 2.1.5.2.1

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{2} x (- x - 4)$ .

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

Step 2.1.5.2.2

Cancel the common factors.

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Step 2.1.5.2.2.1

Factor ${(x - 2)}^{2}$ out of ${(x - 2)}^{6}$ .

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

Step 2.1.5.2.2.2

Cancel the common factor.

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

Step 2.1.5.2.2.3

Rewrite the expression.

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

Step 2.1.5.3

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

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

Step 2.1.5.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.1.5.5

Factor $- 1$ out of $- (x) - 1 (4)$ .

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

Step 2.1.5.6

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

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

Step 2.1.5.7

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 3

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

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Step 3.1

Set the first derivative equal to $0$ .

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

Step 3.2

Set the numerator equal to zero.

$x (x + 4) = 0$

Step 3.3

Solve the equation for $x$ .

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Step 3.3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 4 = 0$

Step 3.3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3.3

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 3.3.3.1

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

$x + 4 = 0$

Step 3.3.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 3.3.4

The final solution is all the values that make $x (x + 4) = 0$ true.

$x = 0, - 4$

Step 4

The values which make the derivative equal to $0$ are $0, - 4$ .

$0, - 4$

Step 5

Find where the derivative is undefined.

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Step 5.1

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

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

Step 5.2

Solve for $x$ .

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Step 5.2.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 5.2.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

$(- \infty, - 4) \cup (- 4, 0) \cup (0, 2) \cup (2, \infty)$

Step 7

Substitute a value from the interval $(- \infty, - 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = - \frac{(- 5) ((- 5) + 4)}{{((- 5) - 2)}^{4}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Add $- 5$ and $4$ .

$f' (- 5) = - \frac{- 5 \cdot - 1}{{(- 5 - 2)}^{4}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

Subtract $2$ from $- 5$ .

$f' (- 5) = - \frac{- 5 \cdot - 1}{{(- 7)}^{4}}$

Step 7.2.2.2

Raise $- 7$ to the power of $4$ .

$f' (- 5) = - \frac{- 5 \cdot - 1}{2401}$

Step 7.2.3

Multiply $- 5$ by $- 1$ .

$f' (- 5) = - \frac{5}{2401}$

Step 7.2.4

The final answer is $- \frac{5}{2401}$ .

$- \frac{5}{2401}$

Step 7.3

At $x = - 5$ the derivative is $- \frac{5}{2401}$ . Since this is negative, the function is decreasing on $(- \infty, - 4)$ .

Decreasing on $(- \infty, - 4)$ since $f' (x) < 0$

Step 8

Substitute a value from the interval $(- 4, 0)$ into the derivative to determine if the function is increasing or decreasing.

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Step 8.1

Replace the variable $x$ with $- 2$ in the expression.

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

Step 8.2

Simplify the result.

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Step 8.2.1

Add $- 2$ and $4$ .

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

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Subtract $2$ from $- 2$ .

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

Step 8.2.2.2

Raise $- 4$ to the power of $4$ .

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

Step 8.2.3

Reduce the expression by cancelling the common factors.

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Step 8.2.3.1

Multiply $- 2$ by $2$ .

$f' (- 2) = - \frac{- 4}{256}$

Step 8.2.3.2

Cancel the common factor of $- 4$ and $256$ .

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Step 8.2.3.2.1

Factor $4$ out of $- 4$ .

$f' (- 2) = - \frac{4 (- 1)}{256}$

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $4$ out of $256$ .

$f' (- 2) = - \frac{4 \cdot - 1}{4 \cdot 64}$

Step 8.2.3.2.2.2

Cancel the common factor.

$f' (- 2) = - \frac{4 \cdot - 1}{4 \cdot 64}$

Step 8.2.3.2.2.3

Rewrite the expression.

$f' (- 2) = - \frac{- 1}{64}$

Step 8.2.3.3

Move the negative in front of the fraction.

$f' (- 2) = \frac{1}{64}$

Step 8.2.4

The final answer is $\frac{1}{64}$ .

$\frac{1}{64}$

Step 8.3

At $x = - 2$ the derivative is $\frac{1}{64}$ . Since this is positive, the function is increasing on $(- 4, 0)$ .

Increasing on $(- 4, 0)$ since $f' (x) > 0$

Step 9

Substitute a value from the interval $(0, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 9.1

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

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

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $1 + 4$ by $1$ .

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

Step 9.2.2

Simplify the denominator.

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Step 9.2.2.1

Subtract $2$ from $1$ .

$f' (1) = - \frac{1 + 4}{{(- 1)}^{4}}$

Step 9.2.2.2

Raise $- 1$ to the power of $4$ .

$f' (1) = - \frac{1 + 4}{1}$

Step 9.2.3

Simplify the expression.

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Step 9.2.3.1

Add $1$ and $4$ .

$f' (1) = - \frac{5}{1}$

Step 9.2.3.2

Divide $5$ by $1$ .

$f' (1) = - 1 \cdot 5$

Step 9.2.3.3

Multiply $- 1$ by $5$ .

$f' (1) = - 5$

Step 9.2.4

The final answer is $- 5$ .

$- 5$

Step 9.3

At $x = 1$ the derivative is $- 5$ . Since this is negative, the function is decreasing on $(0, 2)$ .

Decreasing on $(0, 2)$ since $f' (x) < 0$

Step 10

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 10.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = - \frac{(3) ((3) + 4)}{{((3) - 2)}^{4}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Add $3$ and $4$ .

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

Step 10.2.2

Simplify the denominator.

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Step 10.2.2.1

Subtract $2$ from $3$ .

$f' (3) = - \frac{3 \cdot 7}{1^{4}}$

Step 10.2.2.2

One to any power is one.

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

Step 10.2.3

Simplify the expression.

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Step 10.2.3.1

Multiply $3$ by $7$ .

$f' (3) = - \frac{21}{1}$

Step 10.2.3.2

Divide $21$ by $1$ .

$f' (3) = - 1 \cdot 21$

Step 10.2.3.3

Multiply $- 1$ by $21$ .

$f' (3) = - 21$

Step 10.2.4

The final answer is $- 21$ .

$- 21$

Step 10.3

At $x = 3$ the derivative is $- 21$ . Since this is negative, the function is decreasing on $(2, \infty)$ .

Decreasing on $(2, \infty)$ since $f' (x) < 0$

Step 11

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

Increasing on: $(- 4, 0)$

Decreasing on: $(- \infty, - 4), (0, 2), (2, \infty)$

Step 12