Algebra Examples

$\frac{f {(x - 6)}^{2} + h - x^{2} - 6}{h}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d h} [\frac{f (h)}{g (h)}]$ is $\frac{g (h) \frac{d}{d h} [f (h)] - f (h) \frac{d}{d h} [g (h)]}{{g (h)}^{2}}$ where $f (h) = f {(x - 6)}^{2} + h - x^{2} - 6$ and $g (h) = h$ .

$\frac{h \frac{d}{d h} [f {(x - 6)}^{2} + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 2

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$\frac{h \frac{d}{d h} [f ((x - 6) (x - 6)) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 3

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

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Step 3.1

Apply the distributive property.

$\frac{h \frac{d}{d h} [f (x (x - 6) - 6 (x - 6)) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 3.2

Apply the distributive property.

$\frac{h \frac{d}{d h} [f (x \cdot x + x \cdot - 6 - 6 (x - 6)) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 3.3

Apply the distributive property.

$\frac{h \frac{d}{d h} [f (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

$\frac{h \frac{d}{d h} [f (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 4.1.2

Move $- 6$ to the left of $x$ .

$\frac{h \frac{d}{d h} [f (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{h \frac{d}{d h} [f (x^{2} - 6 x - 6 x + 36) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 4.2

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

$\frac{h \frac{d}{d h} [f (x^{2} - 12 x + 36) + h - x^{2} - 6] - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 5

By the Sum Rule, the derivative of $f (x^{2} - 12 x + 36) + h - x^{2} - 6$ with respect to $h$ is $\frac{d}{d h} [f (x^{2} - 12 x + 36)] + \frac{d}{d h} [h] + \frac{d}{d h} [- x^{2}] + \frac{d}{d h} [- 6]$ .

$\frac{h (\frac{d}{d h} [f (x^{2} - 12 x + 36)] + \frac{d}{d h} [h] + \frac{d}{d h} [- x^{2}] + \frac{d}{d h} [- 6]) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 6

Since $f (x^{2} - 12 x + 36)$ is constant with respect to $h$ , the derivative of $f (x^{2} - 12 x + 36)$ with respect to $h$ is $0$ .

$\frac{h (0 + \frac{d}{d h} [h] + \frac{d}{d h} [- x^{2}] + \frac{d}{d h} [- 6]) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 7

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

$\frac{h (0 + 1 + \frac{d}{d h} [- x^{2}] + \frac{d}{d h} [- 6]) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 8

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

$\frac{h (0 + 1 + 0 + \frac{d}{d h} [- 6]) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 9

Since $- 6$ is constant with respect to $h$ , the derivative of $- 6$ with respect to $h$ is $0$ .

$\frac{h (0 + 1 + 0 + 0) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 10

Combine terms.

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Step 10.1

Add $0$ and $1$ .

$\frac{h (1 + 0 + 0) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 10.2

Add $1$ and $0$ .

$\frac{h (1 + 0) - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 10.3

Add $1$ and $0$ .

$\frac{h \cdot 1 - (f {(x - 6)}^{2} + h - x^{2} - 6) \frac{d}{d h} [h]}{h^{2}}$

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

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

Step 12.2

Apply the distributive property.

$\frac{h \cdot 1 - (f {(x - 6)}^{2}) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3

Simplify the numerator.

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Step 12.3.1

Simplify each term.

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Step 12.3.1.1

Multiply $h$ by $1$ .

$\frac{h - (f {(x - 6)}^{2}) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.2

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$\frac{h - (f ((x - 6) (x - 6))) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.3

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

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Step 12.3.1.3.1

Apply the distributive property.

$\frac{h - (f (x (x - 6) - 6 (x - 6))) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.3.2

Apply the distributive property.

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

Step 12.3.1.3.3

Apply the distributive property.

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

Step 12.3.1.4

Simplify and combine like terms.

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Step 12.3.1.4.1

Simplify each term.

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Step 12.3.1.4.1.1

Multiply $x$ by $x$ .

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

Step 12.3.1.4.1.2

Move $- 6$ to the left of $x$ .

$\frac{h - (f (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6)) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{h - (f (x^{2} - 6 x - 6 x + 36)) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.4.2

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

$\frac{h - (f (x^{2} - 12 x + 36)) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.5

Apply the distributive property.

$\frac{h - (f x^{2} + f (- 12 x) + f \cdot 36) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.6

Simplify.

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Step 12.3.1.6.1

Rewrite using the commutative property of multiplication.

$\frac{h - (f x^{2} - 12 f x + f \cdot 36) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.6.2

Move $36$ to the left of $f$ .

$\frac{h - (f x^{2} - 12 f x + 36 \cdot f) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.7

Apply the distributive property.

$\frac{h + (- (f x^{2}) - (- 12 f x) - (36 f)) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.8

Simplify.

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Step 12.3.1.8.1

Multiply $- 12$ by $- 1$ .

$\frac{h + (- f x^{2} + 12 (f x) - (36 f)) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.8.2

Multiply $36$ by $- 1$ .

$\frac{h + (- f x^{2} + 12 (f x) - 36 f) \cdot 1 - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.9

Multiply $- f x^{2} + 12 f x - 36 f$ by $1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h \cdot 1 - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.10

Multiply $- 1$ by $1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h - - x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.11

Multiply $- - x^{2}$ .

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Step 12.3.1.11.1

Multiply $- 1$ by $- 1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h + 1 x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.11.2

Multiply $x^{2}$ by $1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h + x^{2} \cdot 1 - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.12

Multiply $x^{2}$ by $1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h + x^{2} - - 6 \cdot 1}{h^{2}}$

Step 12.3.1.13

Multiply $- - 6 \cdot 1$ .

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Step 12.3.1.13.1

Multiply $- 1$ by $- 6$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h + x^{2} + 6 \cdot 1}{h^{2}}$

Step 12.3.1.13.2

Multiply $6$ by $1$ .

$\frac{h - f x^{2} + 12 f x - 36 f - h + x^{2} + 6}{h^{2}}$

Step 12.3.2

Combine the opposite terms in $h - f x^{2} + 12 f x - 36 f - h + x^{2} + 6$ .

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Step 12.3.2.1

Subtract $h$ from $h$ .

$\frac{- f x^{2} + 12 f x - 36 f + 0 + x^{2} + 6}{h^{2}}$

Step 12.3.2.2

Add $- f x^{2} + 12 f x - 36 f$ and $0$ .

$\frac{- f x^{2} + 12 f x - 36 f + x^{2} + 6}{h^{2}}$

Step 12.4

Reorder terms.

$\frac{x^{2} - x^{2} f + 12 f x - 36 f + 6}{h^{2}}$