Calculus Examples

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

Step 1

Find the first 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 $f (x) = {(x - 1)}^{2}$ and $g (x) = x - 3$ .

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

Step 1.1.2

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 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Differentiate.

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Step 1.1.3.1

Move $2$ to the left of $x - 3$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Simplify the expression.

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Step 1.1.3.5.1

Add $1$ and $0$ .

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

Step 1.1.3.5.2

Multiply $2$ by $1$ .

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

Step 1.1.3.6

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

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

Step 1.1.3.7

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

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

Step 1.1.3.8

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

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

Step 1.1.3.9

Simplify the expression.

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Step 1.1.3.9.1

Add $1$ and $0$ .

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

Step 1.1.3.9.2

Multiply $- 1$ by $1$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

Apply the distributive property.

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

Step 1.1.4.2

Simplify the numerator.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Multiply $2$ by $- 3$ .

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

Step 1.1.4.2.1.2

Expand $(2 x - 6) (x - 1)$ using the FOIL Method.

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Step 1.1.4.2.1.2.1

Apply the distributive property.

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

Step 1.1.4.2.1.2.2

Apply the distributive property.

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

Step 1.1.4.2.1.2.3

Apply the distributive property.

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

Step 1.1.4.2.1.3

Simplify and combine like terms.

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Step 1.1.4.2.1.3.1

Simplify each term.

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Step 1.1.4.2.1.3.1.1

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

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Step 1.1.4.2.1.3.1.1.1

Move $x$ .

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

Step 1.1.4.2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.4.2.1.3.1.2

Multiply $- 1$ by $2$ .

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

Step 1.1.4.2.1.3.1.3

Multiply $- 6$ by $- 1$ .

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

Step 1.1.4.2.1.3.2

Subtract $6 x$ from $- 2 x$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - {(x - 1)}^{2}}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.4

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

$f' (x) = \frac{2 x^{2} - 8 x + 6 - ((x - 1) (x - 1))}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.5

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

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Step 1.1.4.2.1.5.1

Apply the distributive property.

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x (x - 1) - 1 (x - 1))}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.5.2

Apply the distributive property.

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x \cdot x + x \cdot - 1 - 1 (x - 1))}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.5.3

Apply the distributive property.

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6

Simplify and combine like terms.

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Step 1.1.4.2.1.6.1

Simplify each term.

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Step 1.1.4.2.1.6.1.1

Multiply $x$ by $x$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6.1.2

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

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6.1.3

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

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} - x - 1 x - 1 \cdot - 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6.1.4

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

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} - x - x - 1 \cdot - 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6.1.5

Multiply $- 1$ by $- 1$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} - x - x + 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.6.2

Subtract $x$ from $- x$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - (x^{2} - 2 x + 1)}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.7

Apply the distributive property.

$f' (x) = \frac{2 x^{2} - 8 x + 6 - x^{2} - (- 2 x) - 1 \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.8

Simplify.

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Step 1.1.4.2.1.8.1

Multiply $- 2$ by $- 1$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - x^{2} + 2 x - 1 \cdot 1}{{(x - 3)}^{2}}$

Step 1.1.4.2.1.8.2

Multiply $- 1$ by $1$ .

$f' (x) = \frac{2 x^{2} - 8 x + 6 - x^{2} + 2 x - 1}{{(x - 3)}^{2}}$

Step 1.1.4.2.2

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

$f' (x) = \frac{x^{2} - 8 x + 6 + 2 x - 1}{{(x - 3)}^{2}}$

Step 1.1.4.2.3

Add $- 8 x$ and $2 x$ .

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

Step 1.1.4.2.4

Subtract $1$ from $6$ .

$f' (x) = \frac{x^{2} - 6 x + 5}{{(x - 3)}^{2}}$

Step 1.1.4.3

Factor $x^{2} - 6 x + 5$ using the AC method.

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Step 1.1.4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $5$ and whose sum is $- 6$ .

$- 5, - 1$

Step 1.1.4.3.2

Write the factored form using these integers.

$f' (x) = \frac{(x - 5) (x - 1)}{{(x - 3)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{(x - 5) (x - 1)}{{(x - 3)}^{2}}$ .

$\frac{(x - 5) (x - 1)}{{(x - 3)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{(x - 5) (x - 1)}{{(x - 3)}^{2}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{(x - 5) (x - 1)}{{(x - 3)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$(x - 5) (x - 1) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

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

$x - 5 = 0$

$x - 1 = 0$

Step 2.3.2

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.3.2.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.3.2.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.3.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 2.3.3.1

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

$x - 1 = 0$

Step 2.3.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.3.4

The final solution is all the values that make $(x - 5) (x - 1) = 0$ true.

$x = 5, 1$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Set the denominator in $\frac{(x - 5) (x - 1)}{{(x - 3)}^{2}}$ equal to $0$ to find where the expression is undefined.

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

Step 3.2

Solve for $x$ .

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Step 3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4

Evaluate $\frac{{(x - 1)}^{2}}{x - 3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 5$ .

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Step 4.1.1

Substitute $5$ for $x$ .

$\frac{{((5) - 1)}^{2}}{(5) - 3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

Subtract $1$ from $5$ .

$\frac{4^{2}}{5 - 3}$

Step 4.1.2.1.2

Raise $4$ to the power of $2$ .

$\frac{16}{5 - 3}$

Step 4.1.2.2

Simplify the expression.

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Step 4.1.2.2.1

Subtract $3$ from $5$ .

$\frac{16}{2}$

Step 4.1.2.2.2

Divide $16$ by $2$ .

$8$

Step 4.2

Evaluate at $x = 1$ .

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Step 4.2.1

Substitute $1$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

Subtract $1$ from $1$ .

$\frac{0^{2}}{1 - 3}$

Step 4.2.2.1.2

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

$\frac{0}{1 - 3}$

Step 4.2.2.2

Simplify the expression.

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Step 4.2.2.2.1

Subtract $3$ from $1$ .

$\frac{0}{- 2}$

Step 4.2.2.2.2

Divide $0$ by $- 2$ .

$0$

Step 4.3

Evaluate at $x = 3$ .

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Step 4.3.1

Substitute $3$ for $x$ .

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

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $3$ from $3$ .

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

Step 4.3.2.2

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

Undefined

Step 4.4

List all of the points.

$(5, 8), (1, 0)$

Step 5