Calculus Examples

$f (x) = \frac{x^{2}}{{(x - 1)}^{2}}$ on interval $- 2$ , $- 1$

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 $f (x) = x^{2}$ and $g (x) = {(x - 1)}^{2}$ .

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

Step 1.1.1.2

Differentiate using the Power Rule.

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Step 1.1.1.2.1

Multiply the exponents in ${({(x - 1)}^{2})}^{2}$ .

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Step 1.1.1.2.1.1

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

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

Step 1.1.1.2.1.2

Multiply $2$ by $2$ .

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

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 $n = 2$ .

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

Step 1.1.1.2.3

Move $2$ to the left of ${(x - 1)}^{2}$ .

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

Step 1.1.1.3

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 $g (x) = x - 1$ .

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Step 1.1.1.3.1

To apply the Chain Rule, set $u$ as $x - 1$ .

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

Step 1.1.1.3.2

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

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

Step 1.1.1.3.3

Replace all occurrences of $u$ with $x - 1$ .

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

Step 1.1.1.4

Differentiate.

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Step 1.1.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.1.4.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 1.1.1.4.3

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

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

Step 1.1.1.4.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.1.1.4.5

Simplify the expression.

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Step 1.1.1.4.5.1

Add $1$ and $0$ .

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

Step 1.1.1.4.5.2

Multiply $x - 1$ by $1$ .

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

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Apply the distributive property.

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

Step 1.1.1.5.2

Simplify the numerator.

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Step 1.1.1.5.2.1

Simplify each term.

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Step 1.1.1.5.2.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 1.1.1.5.2.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.1.5.2.1.2.1

Apply the distributive property.

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

Step 1.1.1.5.2.1.2.2

Apply the distributive property.

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

Step 1.1.1.5.2.1.2.3

Apply the distributive property.

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

Step 1.1.1.5.2.1.3

Simplify and combine like terms.

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Step 1.1.1.5.2.1.3.1

Simplify each term.

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Step 1.1.1.5.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.1.5.2.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.1.1.5.2.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.1.5.2.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.1.5.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.1.5.2.1.3.2

Subtract $x$ from $- x$ .

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

Step 1.1.1.5.2.1.4

Apply the distributive property.

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

Step 1.1.1.5.2.1.5

Simplify.

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Step 1.1.1.5.2.1.5.1

Multiply $- 2$ by $2$ .

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

Step 1.1.1.5.2.1.5.2

Multiply $2$ by $1$ .

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

Step 1.1.1.5.2.1.6

Apply the distributive property.

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

Step 1.1.1.5.2.1.7

Simplify.

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Step 1.1.1.5.2.1.7.1

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

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Step 1.1.1.5.2.1.7.1.1

Move $x$ .

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

Step 1.1.1.5.2.1.7.1.2

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

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Step 1.1.1.5.2.1.7.1.2.1

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

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

Step 1.1.1.5.2.1.7.1.2.2

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

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

Step 1.1.1.5.2.1.7.1.3

Add $1$ and $2$ .

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

Step 1.1.1.5.2.1.7.2

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

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Step 1.1.1.5.2.1.7.2.1

Move $x$ .

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

Step 1.1.1.5.2.1.7.2.2

Multiply $x$ by $x$ .

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

Step 1.1.1.5.2.1.8

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

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Step 1.1.1.5.2.1.8.1

Move $x$ .

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

Step 1.1.1.5.2.1.8.2

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

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Step 1.1.1.5.2.1.8.2.1

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

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

Step 1.1.1.5.2.1.8.2.2

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

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

Step 1.1.1.5.2.1.8.3

Add $1$ and $2$ .

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

Step 1.1.1.5.2.1.9

Multiply $- 1$ by $- 2$ .

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

Step 1.1.1.5.2.2

Combine the opposite terms in $2 x^{3} - 4 x^{2} + 2 x - 2 x^{3} + 2 x^{2}$ .

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Step 1.1.1.5.2.2.1

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

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

Step 1.1.1.5.2.2.2

Add $- 4 x^{2} + 2 x$ and $0$ .

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

Step 1.1.1.5.2.3

Add $- 4 x^{2}$ and $2 x^{2}$ .

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

Step 1.1.1.5.3

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

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Step 1.1.1.5.3.1

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

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

Step 1.1.1.5.3.2

Factor $2 x$ out of $2 x$ .

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

Step 1.1.1.5.3.3

Factor $2 x$ out of $2 x (- x) + 2 x (1)$ .

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

Step 1.1.1.5.4

Cancel the common factor of $- x + 1$ and ${(x - 1)}^{4}$ .

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Step 1.1.1.5.4.1

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

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

Step 1.1.1.5.4.2

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

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

Step 1.1.1.5.4.3

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

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

Step 1.1.1.5.4.4

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

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

Step 1.1.1.5.4.5

Factor $x - 1$ out of $2 x (- 1 (x - 1))$ .

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

Step 1.1.1.5.4.6

Cancel the common factors.

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Step 1.1.1.5.4.6.1

Factor $x - 1$ out of ${(x - 1)}^{4}$ .

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

Step 1.1.1.5.4.6.2

Cancel the common factor.

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

Step 1.1.1.5.4.6.3

Rewrite the expression.

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

Step 1.1.1.5.5

Multiply $- 1$ by $2$ .

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

Step 1.1.1.5.6

Move the negative in front of the fraction.

$f' (x) = - \frac{2 x}{{(x - 1)}^{3}}$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{2 x}{{(x - 1)}^{3}}$ .

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

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{2 x}{{(x - 1)}^{3}} = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Set the numerator equal to zero.

$2 x = 0$

Step 1.2.3

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $2$ .

$x = 0$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

Set the denominator in $\frac{2 x}{{(x - 1)}^{3}}$ equal to $0$ to find where the expression is undefined.

${(x - 1)}^{3} = 0$

Step 1.3.2

Solve for $x$ .

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Step 1.3.2.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.3.2.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.4

Evaluate $\frac{x^{2}}{{(x - 1)}^{2}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$\frac{{(0)}^{2}}{{((0) - 1)}^{2}}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{0}{{(0 - 1)}^{2}}$

Step 1.4.1.2.2

Simplify the denominator.

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Step 1.4.1.2.2.1

Subtract $1$ from $0$ .

$\frac{0}{{(- 1)}^{2}}$

Step 1.4.1.2.2.2

Raise $- 1$ to the power of $2$ .

$\frac{0}{1}$

Step 1.4.1.2.3

Divide $0$ by $1$ .

$0$

Step 1.4.2

Evaluate at $x = 1$ .

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Step 1.4.2.1

Substitute $1$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Subtract $1$ from $1$ .

$\frac{{(1)}^{2}}{0^{2}}$

Step 1.4.2.2.2

Raising $0$ to any positive power yields $0$ .

$\frac{{(1)}^{2}}{0}$

Step 1.4.2.2.3

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

Undefined

Step 1.4.3

List all of the points.

$(0, 0)$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = - 2$ .

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Step 3.1.1

Substitute $- 2$ for $x$ .

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

Step 3.1.2

Simplify.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

Subtract $1$ from $- 2$ .

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

Step 3.1.2.2.2

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

$\frac{4}{9}$

Step 3.2

Evaluate at $x = - 1$ .

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Step 3.2.1

Substitute $- 1$ for $x$ .

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $- 1$ to the power of $2$ .

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

Step 3.2.2.2

Simplify the denominator.

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Step 3.2.2.2.1

Subtract $1$ from $- 1$ .

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

Step 3.2.2.2.2

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

$\frac{1}{4}$

Step 3.3

List all of the points.

$(- 2, \frac{4}{9}), (- 1, \frac{1}{4})$

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: $(- 2, \frac{4}{9})$

Absolute Minimum: $(- 1, \frac{1}{4})$

Step 5