Calculus Examples

$y = \frac{x^{2} - 2 a x + a^{2}}{x - a}$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- 2 a$ is constant with respect to $x$ , the derivative of $- 2 a x$ with respect to $x$ is $- 2 a \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $- 2$ by $1$ .

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

Step 2.6

Since $a^{2}$ is constant with respect to $x$ , the derivative of $a^{2}$ with respect to $x$ is $0$ .

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

Step 2.7

Add $2 x - 2 a$ and $0$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

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

Step 2.11.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Apply the distributive property.

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

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

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 3.2.1.2.1.3

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.5

Multiply $- 1$ by $2$ .

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

Step 3.2.1.2.1.6

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.7

Multiply $a$ by $a$ by adding the exponents.

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Step 3.2.1.2.1.7.1

Move $a$ .

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

Step 3.2.1.2.1.7.2

Multiply $a$ by $a$ .

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

Step 3.2.1.2.1.8

Multiply $- 1$ by $- 2$ .

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

Step 3.2.1.2.2

Subtract $2 a x$ from $- 2 x a$ .

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Step 3.2.1.2.2.1

Move $x$ .

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

Step 3.2.1.2.2.2

Subtract $2 a x$ from $- 2 a x$ .

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

Step 3.2.1.3

Multiply $- 2$ by $- 1$ .

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

Step 3.2.2

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

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

Step 3.2.3

Add $- 4 a x$ and $2 a x$ .

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

Step 3.2.4

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

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

Step 3.3

Reorder terms.

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

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

Rearrange terms.

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

Step 3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a x = 2 \cdot a \cdot x$

Step 3.4.3

Rewrite the polynomial.

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

Step 3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = x$ .

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

Step 3.5

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

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Step 3.5.1

Factor $- 1$ out of $a$ .

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

Apply the product rule to $- 1 (- a + x)$ .

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

Step 3.5.5

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

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

Step 3.5.6

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

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

Step 3.5.7

Cancel the common factor.

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

Step 3.5.8

Rewrite the expression.

$1$