Calculus Examples

$lim_{h \to 0} \frac{4 {(x + h - 3)}^{2} - 4 {(x - 3)}^{2}}{h}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{h \to 0} 4 {(x + h - 3)}^{2} - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{lim_{h \to 0} 4 {(x + h - 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.2

Move the term $4$ outside of the limit because it is constant with respect to $h$ .

$\frac{4 lim_{h \to 0} {(x + h - 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.3

Move the exponent $2$ from ${(x + h - 3)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{4 {(lim_{h \to 0} x + h - 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{4 {(lim_{h \to 0} x + lim_{h \to 0} h - lim_{h \to 0} 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.5

Evaluate the limit of $x$ which is constant as $h$ approaches $0$ .

$\frac{4 {(x + lim_{h \to 0} h - lim_{h \to 0} 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.6

Evaluate the limit of $3$ which is constant as $h$ approaches $0$ .

$\frac{4 {(x + lim_{h \to 0} h - 1 \cdot 3)}^{2} - lim_{h \to 0} 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.1.7

Evaluate the limit of $4 {(x - 3)}^{2}$ which is constant as $h$ approaches $0$ .

$\frac{4 {(x + lim_{h \to 0} h - 1 \cdot 3)}^{2} - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{4 {(x + 0 - 1 \cdot 3)}^{2} - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Add $x$ and $0$ .

$\frac{4 {(x - 1 \cdot 3)}^{2} - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2

Simplify each term.

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Step 1.1.2.3.2.1

Multiply $- 1$ by $3$ .

$\frac{4 {(x - 3)}^{2} - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.2

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{4 ((x - 3) (x - 3)) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.3

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.2.3.2.3.1

Apply the distributive property.

$\frac{4 (x (x - 3) - 3 (x - 3)) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.3.2

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot - 3 - 3 (x - 3)) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.3.3

Apply the distributive property.

$\frac{4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.4

Simplify and combine like terms.

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Step 1.1.2.3.2.4.1

Simplify each term.

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Step 1.1.2.3.2.4.1.1

Multiply $x$ by $x$ .

$\frac{4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.4.1.2

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

$\frac{4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.4.1.3

Multiply $- 3$ by $- 3$ .

$\frac{4 (x^{2} - 3 x - 3 x + 9) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.4.2

Subtract $3 x$ from $- 3 x$ .

$\frac{4 (x^{2} - 6 x + 9) - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.5

Apply the distributive property.

$\frac{4 x^{2} + 4 (- 6 x) + 4 \cdot 9 - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.6

Simplify.

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Step 1.1.2.3.2.6.1

Multiply $- 6$ by $4$ .

$\frac{4 x^{2} - 24 x + 4 \cdot 9 - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.6.2

Multiply $4$ by $9$ .

$\frac{4 x^{2} - 24 x + 36 - 4 {(x - 3)}^{2}}{lim_{h \to 0} h}$

Step 1.1.2.3.2.7

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$\frac{4 x^{2} - 24 x + 36 - 4 ((x - 3) (x - 3))}{lim_{h \to 0} h}$

Step 1.1.2.3.2.8

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.2.3.2.8.1

Apply the distributive property.

$\frac{4 x^{2} - 24 x + 36 - 4 (x (x - 3) - 3 (x - 3))}{lim_{h \to 0} h}$

Step 1.1.2.3.2.8.2

Apply the distributive property.

$\frac{4 x^{2} - 24 x + 36 - 4 (x \cdot x + x \cdot - 3 - 3 (x - 3))}{lim_{h \to 0} h}$

Step 1.1.2.3.2.8.3

Apply the distributive property.

$\frac{4 x^{2} - 24 x + 36 - 4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)}{lim_{h \to 0} h}$

Step 1.1.2.3.2.9

Simplify and combine like terms.

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Step 1.1.2.3.2.9.1

Simplify each term.

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Step 1.1.2.3.2.9.1.1

Multiply $x$ by $x$ .

$\frac{4 x^{2} - 24 x + 36 - 4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)}{lim_{h \to 0} h}$

Step 1.1.2.3.2.9.1.2

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

$\frac{4 x^{2} - 24 x + 36 - 4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)}{lim_{h \to 0} h}$

Step 1.1.2.3.2.9.1.3

Multiply $- 3$ by $- 3$ .

$\frac{4 x^{2} - 24 x + 36 - 4 (x^{2} - 3 x - 3 x + 9)}{lim_{h \to 0} h}$

Step 1.1.2.3.2.9.2

Subtract $3 x$ from $- 3 x$ .

$\frac{4 x^{2} - 24 x + 36 - 4 (x^{2} - 6 x + 9)}{lim_{h \to 0} h}$

Step 1.1.2.3.2.10

Apply the distributive property.

$\frac{4 x^{2} - 24 x + 36 - 4 x^{2} - 4 (- 6 x) - 4 \cdot 9}{lim_{h \to 0} h}$

Step 1.1.2.3.2.11

Simplify.

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Step 1.1.2.3.2.11.1

Multiply $- 6$ by $- 4$ .

$\frac{4 x^{2} - 24 x + 36 - 4 x^{2} + 24 x - 4 \cdot 9}{lim_{h \to 0} h}$

Step 1.1.2.3.2.11.2

Multiply $- 4$ by $9$ .

$\frac{4 x^{2} - 24 x + 36 - 4 x^{2} + 24 x - 36}{lim_{h \to 0} h}$

Step 1.1.2.3.3

Combine the opposite terms in $4 x^{2} - 24 x + 36 - 4 x^{2} + 24 x - 36$ .

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Step 1.1.2.3.3.1

Subtract $4 x^{2}$ from $4 x^{2}$ .

$\frac{- 24 x + 36 + 0 + 24 x - 36}{lim_{h \to 0} h}$

Step 1.1.2.3.3.2

Add $- 24 x + 36$ and $0$ .

$\frac{- 24 x + 36 + 24 x - 36}{lim_{h \to 0} h}$

Step 1.1.2.3.3.3

Add $- 24 x$ and $24 x$ .

$\frac{0 + 36 - 36}{lim_{h \to 0} h}$

Step 1.1.2.3.3.4

Add $0$ and $36$ .

$\frac{36 - 36}{lim_{h \to 0} h}$

Step 1.1.2.3.3.5

Subtract $36$ from $36$ .

$\frac{0}{lim_{h \to 0} h}$

Step 1.1.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{h \to 0} \frac{4 {(x + h - 3)}^{2} - 4 {(x - 3)}^{2}}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [4 {(x + h - 3)}^{2} - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [4 {(x + h - 3)}^{2} - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.2

Rewrite ${(x + h - 3)}^{2}$ as $(x + h - 3) (x + h - 3)$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 ((x + h - 3) (x + h - 3)) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.3

Expand $(x + h - 3) (x + h - 3)$ by multiplying each term in the first expression by each term in the second expression.

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x \cdot x + x h + x \cdot - 3 + h x + h \cdot h + h \cdot - 3 - 3 x - 3 h - 3 \cdot - 3) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4

Simplify each term.

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Step 1.3.4.1

Multiply $x$ by $x$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + x h + x \cdot - 3 + h x + h \cdot h + h \cdot - 3 - 3 x - 3 h - 3 \cdot - 3) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.2

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

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + x h - 3 \cdot x + h x + h \cdot h + h \cdot - 3 - 3 x - 3 h - 3 \cdot - 3) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.3

Multiply $h$ by $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + x h - 3 x + h x + h^{2} + h \cdot - 3 - 3 x - 3 h - 3 \cdot - 3) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.4

Move $- 3$ to the left of $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + x h - 3 x + h x + h^{2} - 3 \cdot h - 3 x - 3 h - 3 \cdot - 3) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.4.5

Multiply $- 3$ by $- 3$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + x h - 3 x + h x + h^{2} - 3 h - 3 x - 3 h + 9) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.5

Add $x h$ and $h x$ .

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Step 1.3.5.1

Reorder $x$ and $h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + h x + h x - 3 x + h^{2} - 3 h - 3 x - 3 h + 9) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.5.2

Add $h x$ and $h x$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x - 3 x + h^{2} - 3 h - 3 x - 3 h + 9) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.6

Subtract $3 x$ from $- 3 x$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 3 h - 6 x - 3 h + 9) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.7

Subtract $3 h$ from $- 3 h$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 {(x - 3)}^{2}]}{\frac{d}{d h} [h]}$

Step 1.3.8

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 ((x - 3) (x - 3))]}{\frac{d}{d h} [h]}$

Step 1.3.9

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.3.9.1

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x (x - 3) - 3 (x - 3))]}{\frac{d}{d h} [h]}$

Step 1.3.9.2

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x \cdot x + x \cdot - 3 - 3 (x - 3))]}{\frac{d}{d h} [h]}$

Step 1.3.9.3

Apply the distributive property.

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)]}{\frac{d}{d h} [h]}$

Step 1.3.10

Simplify and combine like terms.

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Step 1.3.10.1

Simplify each term.

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Step 1.3.10.1.1

Multiply $x$ by $x$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)]}{\frac{d}{d h} [h]}$

Step 1.3.10.1.2

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

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)]}{\frac{d}{d h} [h]}$

Step 1.3.10.1.3

Multiply $- 3$ by $- 3$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x^{2} - 3 x - 3 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.10.2

Subtract $3 x$ from $- 3 x$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.11

By the Sum Rule, the derivative of $4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9) - 4 (x^{2} - 6 x + 9)$ with respect to $h$ is $\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9)] + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9)] + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12

Evaluate $\frac{d}{d h} [4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9)]$ .

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Step 1.3.12.1

Since $4$ is constant with respect to $h$ , the derivative of $4 (x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9)$ with respect to $h$ is $4 \frac{d}{d h} [x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9]$ .

$lim_{h \to 0} \frac{4 \frac{d}{d h} [x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9] + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.2

By the Sum Rule, the derivative of $x^{2} + 2 h x + h^{2} - 6 h - 6 x + 9$ with respect to $h$ is $\frac{d}{d h} [x^{2}] + \frac{d}{d h} [2 h x] + \frac{d}{d h} [h^{2}] + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]$ .

$lim_{h \to 0} \frac{4 (\frac{d}{d h} [x^{2}] + \frac{d}{d h} [2 h x] + \frac{d}{d h} [h^{2}] + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.3

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

$lim_{h \to 0} \frac{4 (0 + \frac{d}{d h} [2 h x] + \frac{d}{d h} [h^{2}] + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.4

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

$lim_{h \to 0} \frac{4 (0 + 2 x \frac{d}{d h} [h] + \frac{d}{d h} [h^{2}] + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.5

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

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + \frac{d}{d h} [h^{2}] + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.6

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

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + 2 h + \frac{d}{d h} [- 6 h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.7

Since $- 6$ is constant with respect to $h$ , the derivative of $- 6 h$ with respect to $h$ is $- 6 \frac{d}{d h} [h]$ .

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + 2 h - 6 \frac{d}{d h} [h] + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.8

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

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + 2 h - 6 \cdot 1 + \frac{d}{d h} [- 6 x] + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.9

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

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + 2 h - 6 \cdot 1 + 0 + \frac{d}{d h} [9]) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.10

Since $9$ is constant with respect to $h$ , the derivative of $9$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{4 (0 + 2 x \cdot 1 + 2 h - 6 \cdot 1 + 0 + 0) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.11

Multiply $2$ by $1$ .

$lim_{h \to 0} \frac{4 (0 + 2 x + 2 h - 6 \cdot 1 + 0 + 0) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.12

Add $0$ and $2 x$ .

$lim_{h \to 0} \frac{4 (2 x + 2 h - 6 \cdot 1 + 0 + 0) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.13

Multiply $- 6$ by $1$ .

$lim_{h \to 0} \frac{4 (2 x + 2 h - 6 + 0 + 0) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.14

Add $2 x + 2 h - 6$ and $0$ .

$lim_{h \to 0} \frac{4 (2 x + 2 h - 6 + 0) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.12.15

Add $2 x + 2 h - 6$ and $0$ .

$lim_{h \to 0} \frac{4 (2 x + 2 h - 6) + \frac{d}{d h} [- 4 (x^{2} - 6 x + 9)]}{\frac{d}{d h} [h]}$

Step 1.3.13

Since $- 4 (x^{2} - 6 x + 9)$ is constant with respect to $h$ , the derivative of $- 4 (x^{2} - 6 x + 9)$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{4 (2 x + 2 h - 6) + 0}{\frac{d}{d h} [h]}$

Step 1.3.14

Simplify.

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Step 1.3.14.1

Apply the distributive property.

$lim_{h \to 0} \frac{4 (2 x) + 4 (2 h) + 4 \cdot - 6 + 0}{\frac{d}{d h} [h]}$

Step 1.3.14.2

Combine terms.

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Step 1.3.14.2.1

Multiply $2$ by $4$ .

$lim_{h \to 0} \frac{8 x + 4 (2 h) + 4 \cdot - 6 + 0}{\frac{d}{d h} [h]}$

Step 1.3.14.2.2

Multiply $2$ by $4$ .

$lim_{h \to 0} \frac{8 x + 8 h + 4 \cdot - 6 + 0}{\frac{d}{d h} [h]}$

Step 1.3.14.2.3

Multiply $4$ by $- 6$ .

$lim_{h \to 0} \frac{8 x + 8 h - 24 + 0}{\frac{d}{d h} [h]}$

Step 1.3.14.2.4

Add $8 x + 8 h - 24$ and $0$ .

$lim_{h \to 0} \frac{8 x + 8 h - 24}{\frac{d}{d h} [h]}$

Step 1.3.14.3

Reorder terms.

$lim_{h \to 0} \frac{8 h + 8 x - 24}{\frac{d}{d h} [h]}$

Step 1.3.15

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

$lim_{h \to 0} \frac{8 h + 8 x - 24}{1}$

Step 1.4

Divide $8 h + 8 x - 24$ by $1$ .

$lim_{h \to 0} 8 h + 8 x - 24$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$lim_{h \to 0} 8 h + lim_{h \to 0} 8 x - lim_{h \to 0} 24$

Step 2.2

Move the term $8$ outside of the limit because it is constant with respect to $h$ .

$8 lim_{h \to 0} h + lim_{h \to 0} 8 x - lim_{h \to 0} 24$

Step 2.3

Evaluate the limit of $8 x$ which is constant as $h$ approaches $0$ .

$8 lim_{h \to 0} h + 8 x - lim_{h \to 0} 24$

Step 2.4

Evaluate the limit of $24$ which is constant as $h$ approaches $0$ .

$8 lim_{h \to 0} h + 8 x - 1 \cdot 24$

Step 3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$8 \cdot 0 + 8 x - 1 \cdot 24$

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $8$ by $0$ .

$0 + 8 x - 1 \cdot 24$

Step 4.1.2

Multiply $- 1$ by $24$ .

$0 + 8 x - 24$

Step 4.2

Add $0$ and $8 x$ .

$8 x - 24$