Calculus Examples

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

Step 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 $g (x) = x - 5$ .

$\frac{(x - 5) \frac{d}{d x} [{(x - 1)}^{2}] - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 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 2.1

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

$\frac{(x - 5) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x - 1]) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.2

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

$\frac{(x - 5) (2 u \frac{d}{d x} [x - 1]) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.3

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

$\frac{(x - 5) (2 (x - 1) \frac{d}{d x} [x - 1]) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3

Differentiate.

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Step 3.1

Move $2$ to the left of $x - 5$ .

$\frac{2 \cdot (x - 5) (x - 1) \frac{d}{d x} [x - 1] - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.2

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

$\frac{2 (x - 5) (x - 1) (\frac{d}{d x} [x] + \frac{d}{d x} [- 1]) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.3

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

$\frac{2 (x - 5) (x - 1) (1 + \frac{d}{d x} [- 1]) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.4

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

$\frac{2 (x - 5) (x - 1) (1 + 0) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$\frac{2 (x - 5) (x - 1) \cdot 1 - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.5.2

Multiply $2$ by $1$ .

$\frac{2 (x - 5) (x - 1) - {(x - 1)}^{2} \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 3.6

By the Sum Rule, the derivative of $x - 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 5]$ .

$\frac{2 (x - 5) (x - 1) - {(x - 1)}^{2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 3.7

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

$\frac{2 (x - 5) (x - 1) - {(x - 1)}^{2} (1 + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 3.8

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

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

Step 3.9

Simplify the expression.

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Step 3.9.1

Add $1$ and $0$ .

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

Step 3.9.2

Multiply $- 1$ by $1$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by $- 5$ .

$\frac{(2 x - 10) (x - 1) - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.2

Expand $(2 x - 10) (x - 1)$ using the FOIL Method.

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Step 4.2.1.2.1

Apply the distributive property.

$\frac{2 x (x - 1) - 10 (x - 1) - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.2.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 1 - 10 (x - 1) - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.2.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 1 - 10 x - 10 \cdot - 1 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.3

Simplify and combine like terms.

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Step 4.2.1.3.1

Simplify each term.

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Step 4.2.1.3.1.1

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

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Step 4.2.1.3.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 1 - 10 x - 10 \cdot - 1 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.3.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 1 - 10 x - 10 \cdot - 1 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.3.1.2

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} - 2 x - 10 x - 10 \cdot - 1 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.3.1.3

Multiply $- 10$ by $- 1$ .

$\frac{2 x^{2} - 2 x - 10 x + 10 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.3.2

Subtract $10 x$ from $- 2 x$ .

$\frac{2 x^{2} - 12 x + 10 - {(x - 1)}^{2}}{{(x - 5)}^{2}}$

Step 4.2.1.4

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

$\frac{2 x^{2} - 12 x + 10 - ((x - 1) (x - 1))}{{(x - 5)}^{2}}$

Step 4.2.1.5

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

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Step 4.2.1.5.1

Apply the distributive property.

$\frac{2 x^{2} - 12 x + 10 - (x (x - 1) - 1 (x - 1))}{{(x - 5)}^{2}}$

Step 4.2.1.5.2

Apply the distributive property.

$\frac{2 x^{2} - 12 x + 10 - (x \cdot x + x \cdot - 1 - 1 (x - 1))}{{(x - 5)}^{2}}$

Step 4.2.1.5.3

Apply the distributive property.

$\frac{2 x^{2} - 12 x + 10 - (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6

Simplify and combine like terms.

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Step 4.2.1.6.1

Simplify each term.

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Step 4.2.1.6.1.1

Multiply $x$ by $x$ .

$\frac{2 x^{2} - 12 x + 10 - (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6.1.2

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

$\frac{2 x^{2} - 12 x + 10 - (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6.1.3

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

$\frac{2 x^{2} - 12 x + 10 - (x^{2} - x - 1 x - 1 \cdot - 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6.1.4

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

$\frac{2 x^{2} - 12 x + 10 - (x^{2} - x - x - 1 \cdot - 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6.1.5

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} - 12 x + 10 - (x^{2} - x - x + 1)}{{(x - 5)}^{2}}$

Step 4.2.1.6.2

Subtract $x$ from $- x$ .

$\frac{2 x^{2} - 12 x + 10 - (x^{2} - 2 x + 1)}{{(x - 5)}^{2}}$

Step 4.2.1.7

Apply the distributive property.

$\frac{2 x^{2} - 12 x + 10 - x^{2} - (- 2 x) - 1 \cdot 1}{{(x - 5)}^{2}}$

Step 4.2.1.8

Simplify.

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Step 4.2.1.8.1

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{2} - 12 x + 10 - x^{2} + 2 x - 1 \cdot 1}{{(x - 5)}^{2}}$

Step 4.2.1.8.2

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} - 12 x + 10 - x^{2} + 2 x - 1}{{(x - 5)}^{2}}$

Step 4.2.2

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

$\frac{x^{2} - 12 x + 10 + 2 x - 1}{{(x - 5)}^{2}}$

Step 4.2.3

Add $- 12 x$ and $2 x$ .

$\frac{x^{2} - 10 x + 10 - 1}{{(x - 5)}^{2}}$

Step 4.2.4

Subtract $1$ from $10$ .

$\frac{x^{2} - 10 x + 9}{{(x - 5)}^{2}}$

Step 4.3

Factor $x^{2} - 10 x + 9$ using the AC method.

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Step 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 $9$ and whose sum is $- 10$ .

$- 9, - 1$

Step 4.3.2

Write the factored form using these integers.

$\frac{(x - 9) (x - 1)}{{(x - 5)}^{2}}$