Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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

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

Step 1.1.2

Differentiate using the Power Rule.

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Step 1.1.2.1

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

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 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 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.4

Differentiate.

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Step 1.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.4.2

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

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

Step 1.1.4.3

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

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

Step 1.1.4.4

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

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

Step 1.1.4.5

Simplify the expression.

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Step 1.1.4.5.1

Add $1$ and $0$ .

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

Step 1.1.4.5.2

Multiply $x - 1$ by $1$ .

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

Step 1.1.5

Simplify.

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Step 1.1.5.1

Apply the distributive property.

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

Step 1.1.5.2

Simplify the numerator.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

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

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

Step 1.1.5.2.1.2

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

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Step 1.1.5.2.1.2.1

Apply the distributive property.

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

Step 1.1.5.2.1.2.2

Apply the distributive property.

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

Step 1.1.5.2.1.2.3

Apply the distributive property.

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

Step 1.1.5.2.1.3

Simplify and combine like terms.

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Step 1.1.5.2.1.3.1

Simplify each term.

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Step 1.1.5.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.5.2.1.3.1.2

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

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

Step 1.1.5.2.1.3.1.3

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

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

Step 1.1.5.2.1.3.1.4

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

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

Step 1.1.5.2.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.5.2.1.3.2

Subtract $x$ from $- x$ .

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

Step 1.1.5.2.1.4

Apply the distributive property.

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

Step 1.1.5.2.1.5

Simplify.

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Step 1.1.5.2.1.5.1

Multiply $- 2$ by $2$ .

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

Step 1.1.5.2.1.5.2

Multiply $2$ by $1$ .

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

Step 1.1.5.2.1.6

Apply the distributive property.

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

Step 1.1.5.2.1.7

Simplify.

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Step 1.1.5.2.1.7.1

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

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Step 1.1.5.2.1.7.1.1

Move $x$ .

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

Step 1.1.5.2.1.7.1.2

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

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Step 1.1.5.2.1.7.1.2.1

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

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

Step 1.1.5.2.1.7.1.2.2

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

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

Step 1.1.5.2.1.7.1.3

Add $1$ and $2$ .

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

Step 1.1.5.2.1.7.2

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

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Step 1.1.5.2.1.7.2.1

Move $x$ .

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

Step 1.1.5.2.1.7.2.2

Multiply $x$ by $x$ .

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

Step 1.1.5.2.1.8

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

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Step 1.1.5.2.1.8.1

Move $x$ .

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

Step 1.1.5.2.1.8.2

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

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Step 1.1.5.2.1.8.2.1

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

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

Step 1.1.5.2.1.8.2.2

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

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

Step 1.1.5.2.1.8.3

Add $1$ and $2$ .

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

Step 1.1.5.2.1.9

Multiply $- 1$ by $- 2$ .

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

Step 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 1.1.5.2.2.1

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

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

Step 1.1.5.2.2.2

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

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

Step 1.1.5.2.3

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

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

Step 1.1.5.3

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

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Step 1.1.5.3.1

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

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

Step 1.1.5.3.2

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

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

Step 1.1.5.3.3

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

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

Step 1.1.5.4

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

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Step 1.1.5.4.1

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

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

Step 1.1.5.4.2

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

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

Step 1.1.5.4.3

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

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

Step 1.1.5.4.4

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

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

Step 1.1.5.4.5

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

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

Step 1.1.5.4.6

Cancel the common factors.

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Step 1.1.5.4.6.1

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

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

Step 1.1.5.4.6.2

Cancel the common factor.

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

Step 1.1.5.4.6.3

Rewrite the expression.

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

Step 1.1.5.5

Multiply $- 1$ by $2$ .

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

Step 1.1.5.6

Move the negative in front of the fraction.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2 x}{{(x - 1)}^{3}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{x}{{(x - 1)}^{3}}]$ .

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

Step 1.2.2

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

Step 1.2.3

Differentiate using the Power Rule.

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Step 1.2.3.1

Multiply the exponents in ${({(x - 1)}^{3})}^{2}$ .

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Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

Multiply $3$ by $2$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{3} \frac{d}{d x} (x) - x \frac{d}{d x} {(x - 1)}^{3}}{{(x - 1)}^{6}}$

Step 1.2.3.2

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

$f'' (x) = - 2 \frac{{(x - 1)}^{3} \cdot 1 - x \frac{d}{d x} {(x - 1)}^{3}}{{(x - 1)}^{6}}$

Step 1.2.3.3

Multiply ${(x - 1)}^{3}$ by $1$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{3} - x \frac{d}{d x} {(x - 1)}^{3}}{{(x - 1)}^{6}}$

Step 1.2.4

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 $g (x) = x - 1$ .

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Step 1.2.4.1

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

$f'' (x) = - 2 \frac{{(x - 1)}^{3} - x (\frac{d}{d u} (u^{3}) \frac{d}{d x} (x - 1))}{{(x - 1)}^{6}}$

Step 1.2.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{3} - x (3 u^{2} \frac{d}{d x} (x - 1))}{{(x - 1)}^{6}}$

Step 1.2.4.3

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

$f'' (x) = - 2 \frac{{(x - 1)}^{3} - x (3 {(x - 1)}^{2} \frac{d}{d x} (x - 1))}{{(x - 1)}^{6}}$

Step 1.2.5

Simplify with factoring out.

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Step 1.2.5.1

Multiply $3$ by $- 1$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{3} - 3 x ({(x - 1)}^{2} \frac{d}{d x} (x - 1))}{{(x - 1)}^{6}}$

Step 1.2.5.2

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{3} - 3 x ({(x - 1)}^{2} \frac{d}{d x} [x - 1])$ .

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Step 1.2.5.2.1

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{3}$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{2} (x - 1) - 3 x ({(x - 1)}^{2} \frac{d}{d x} (x - 1))}{{(x - 1)}^{6}}$

Step 1.2.5.2.2

Factor ${(x - 1)}^{2}$ out of $- 3 x ({(x - 1)}^{2} \frac{d}{d x} [x - 1])$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{2} (x - 1) + {(x - 1)}^{2} (- 3 x (\frac{d}{d x} (x - 1)))}{{(x - 1)}^{6}}$

Step 1.2.5.2.3

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{2} (x - 1) + {(x - 1)}^{2} (- 3 x (\frac{d}{d x} [x - 1]))$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{2} (x - 1 - 3 x (\frac{d}{d x} (x - 1)))}{{(x - 1)}^{6}}$

Step 1.2.6

Cancel the common factors.

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Step 1.2.6.1

Factor ${(x - 1)}^{2}$ out of ${(x - 1)}^{6}$ .

$f'' (x) = - 2 \frac{{(x - 1)}^{2} (x - 1 - 3 x (\frac{d}{d x} (x - 1)))}{{(x - 1)}^{2} {(x - 1)}^{4}}$

Step 1.2.6.2

Cancel the common factor.

$f'' (x) = - 2 \frac{{(x - 1)}^{2} (x - 1 - 3 x (\frac{d}{d x} (x - 1)))}{{(x - 1)}^{2} {(x - 1)}^{4}}$

Step 1.2.6.3

Rewrite the expression.

$f'' (x) = - 2 \frac{x - 1 - 3 x (\frac{d}{d x} (x - 1))}{{(x - 1)}^{4}}$

Step 1.2.7

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

$f'' (x) = - 2 \frac{x - 1 - 3 x (\frac{d}{d x} (x) + \frac{d}{d x} (- 1))}{{(x - 1)}^{4}}$

Step 1.2.8

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

$f'' (x) = - 2 \frac{x - 1 - 3 x (1 + \frac{d}{d x} (- 1))}{{(x - 1)}^{4}}$

Step 1.2.9

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

$f'' (x) = - 2 \frac{x - 1 - 3 x (1 + 0)}{{(x - 1)}^{4}}$

Step 1.2.10

Simplify terms.

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Step 1.2.10.1

Add $1$ and $0$ .

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

Step 1.2.10.2

Multiply $- 3$ by $1$ .

$f'' (x) = - 2 \frac{x - 1 - 3 x}{{(x - 1)}^{4}}$

Step 1.2.10.3

Subtract $3 x$ from $x$ .

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

Step 1.2.10.4

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

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

Step 1.2.10.5

Move the negative in front of the fraction.

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

Step 1.2.11

Simplify.

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Step 1.2.11.1

Apply the distributive property.

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

Step 1.2.11.2

Simplify each term.

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Step 1.2.11.2.1

Multiply $- 2$ by $2$ .

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

Step 1.2.11.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.11.3

Factor $2$ out of $- 4 x - 2$ .

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Step 1.2.11.3.1

Factor $2$ out of $- 4 x$ .

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

Step 1.2.11.3.2

Factor $2$ out of $- 2$ .

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

Step 1.2.11.3.3

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

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

Step 1.2.11.4

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

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

Step 1.2.11.5

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

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

Step 1.2.11.6

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

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

Step 1.2.11.7

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

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

Step 1.2.11.8

Move the negative in front of the fraction.

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

Step 1.2.11.9

Multiply $- 1$ by $- 1$ .

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

Step 1.2.11.10

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

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2 (2 x + 1)}{{(x - 1)}^{4}}$ .

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

Step 2

Set the second derivative equal to $0$ then solve the equation $\frac{2 (2 x + 1)}{{(x - 1)}^{4}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$2 (2 x + 1) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $2 (2 x + 1) = 0$ by $2$ and simplify.

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Step 2.3.1.1

Divide each term in $2 (2 x + 1) = 0$ by $2$ .

$\frac{2 (2 x + 1)}{2} = \frac{0}{2}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $2$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{2 (2 x + 1)}{2} = \frac{0}{2}$

Step 2.3.1.2.1.2

Divide $2 x + 1$ by $1$ .

$2 x + 1 = \frac{0}{2}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $2$ .

$2 x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.3.3

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 2.3.3.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $- \frac{1}{2}$ in $f (x) = \frac{x^{2}}{{(x - 1)}^{2}}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify the numerator.

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Step 3.1.2.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 3.1.2.1.2

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

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

Step 3.1.2.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 3.1.2.1.4

One to any power is one.

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

Step 3.1.2.1.5

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

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

Step 3.1.2.1.6

Multiply $\frac{1}{4}$ by $1$ .

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

Step 3.1.2.2

Simplify the denominator.

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Step 3.1.2.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.1.2.2.2

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 3.1.2.2.3

Combine the numerators over the common denominator.

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

Step 3.1.2.2.4

Simplify the numerator.

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Step 3.1.2.2.4.1

Multiply $- 1$ by $2$ .

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

Step 3.1.2.2.4.2

Subtract $2$ from $- 1$ .

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{{(\frac{- 3}{2})}^{2}}$

Step 3.1.2.2.5

Move the negative in front of the fraction.

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{{(- \frac{3}{2})}^{2}}$

Step 3.1.2.2.6

Apply the product rule to $- \frac{3}{2}$ .

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{{(- 1)}^{2} {(\frac{3}{2})}^{2}}$

Step 3.1.2.2.7

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

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{1 {(\frac{3}{2})}^{2}}$

Step 3.1.2.2.8

Apply the product rule to $\frac{3}{2}$ .

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{1 (\frac{3^{2}}{2^{2}})}$

Step 3.1.2.2.9

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

$f (- \frac{1}{2}) = \frac{\frac{1}{4}}{1 (\frac{9}{2^{2}})}$

Step 3.1.2.2.10

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

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

Step 3.1.2.2.11

Multiply $\frac{9}{4}$ by $1$ .

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

Step 3.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{1}{2}) = \frac{1}{4} \cdot \frac{4}{9}$

Step 3.1.2.4

Cancel the common factor of $4$ .

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Step 3.1.2.4.1

Cancel the common factor.

$f (- \frac{1}{2}) = \frac{1}{4} \cdot \frac{4}{9}$

Step 3.1.2.4.2

Rewrite the expression.

$f (- \frac{1}{2}) = \frac{1}{9}$

Step 3.1.2.5

The final answer is $\frac{1}{9}$ .

$\frac{1}{9}$

Step 3.2

The point found by substituting $- \frac{1}{2}$ in $f (x) = \frac{x^{2}}{{(x - 1)}^{2}}$ is $(- \frac{1}{2}, \frac{1}{9})$ . This point can be an inflection point.

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

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - \frac{1}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 0.6$ in the expression.

$f'' (- 0.6) = \frac{2 (2 (- 0.6) + 1)}{{((- 0.6) - 1)}^{4}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $2$ by $- 0.6$ .

$f'' (- 0.6) = \frac{2 (- 1.2 + 1)}{{(- 0.6 - 1)}^{4}}$

Step 5.2.1.2

Add $- 1.2$ and $1$ .

$f'' (- 0.6) = \frac{2 \cdot - 0.2}{{(- 0.6 - 1)}^{4}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Subtract $1$ from $- 0.6$ .

$f'' (- 0.6) = \frac{2 \cdot - 0.2}{{(- 1.6)}^{4}}$

Step 5.2.2.2

Raise $- 1.6$ to the power of $4$ .

$f'' (- 0.6) = \frac{2 \cdot - 0.2}{6.5536}$

Step 5.2.3

Simplify the expression.

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Step 5.2.3.1

Multiply $2$ by $- 0.2$ .

$f'' (- 0.6) = \frac{- 0.4}{6.5536}$

Step 5.2.3.2

Divide $- 0.4$ by $6.5536$ .

$f'' (- 0.6) = - 0.06103515$

Step 5.2.4

The final answer is $- 0.06103515$ .

$- 0.06103515$

Step 5.3

At $- 0.6$ , the second derivative is $- 0.06103515$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - \frac{1}{2})$

Decreasing on $(- \infty, - \frac{1}{2})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- \frac{1}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 0.4$ in the expression.

$f'' (- 0.4) = \frac{2 (2 (- 0.4) + 1)}{{((- 0.4) - 1)}^{4}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $2$ by $- 0.4$ .

$f'' (- 0.4) = \frac{2 (- 0.8 + 1)}{{(- 0.4 - 1)}^{4}}$

Step 6.2.1.2

Add $- 0.8$ and $1$ .

$f'' (- 0.4) = \frac{2 \cdot 0.2}{{(- 0.4 - 1)}^{4}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Subtract $1$ from $- 0.4$ .

$f'' (- 0.4) = \frac{2 \cdot 0.2}{{(- 1.4)}^{4}}$

Step 6.2.2.2

Raise $- 1.4$ to the power of $4$ .

$f'' (- 0.4) = \frac{2 \cdot 0.2}{3.8416}$

Step 6.2.3

Simplify the expression.

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Step 6.2.3.1

Multiply $2$ by $0.2$ .

$f'' (- 0.4) = \frac{0.4}{3.8416}$

Step 6.2.3.2

Divide $0.4$ by $3.8416$ .

$f'' (- 0.4) = 0.10412328$

Step 6.2.4

The final answer is $0.10412328$ .

$0.10412328$

Step 6.3

At $- 0.4$ , the second derivative is $0.10412328$ . Since this is positive, the second derivative is increasing on the interval $(- \frac{1}{2}, \infty)$ .

Increasing on $(- \frac{1}{2}, \infty)$ since $f'' (x) > 0$

Step 7

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection point in this case is $(- \frac{1}{2}, \frac{1}{9})$ .

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

Step 8