Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

To write $\frac{1}{{(x + h)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.2

To write $- \frac{1}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 1.3

Write each expression with a common denominator of ${(x + h)}^{2} x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{{(x + h)}^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Step 1.3.2

Multiply $\frac{1}{x^{2}}$ by $\frac{{(x + h)}^{2}}{{(x + h)}^{2}}$ .

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

Step 1.3.3

Reorder the factors of ${(x + h)}^{2} x^{2}$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

Multiply $\frac{x^{2} - {(x + h)}^{2}}{x^{2} {(x + h)}^{2}}$ by $\frac{1}{h}$ .

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

Step 2.2

Move the term $\frac{1}{x^{2}}$ outside of the limit because it is constant with respect to $h$ .

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

Evaluate the limit.

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

Evaluate the limit of $x^{2}$ which is constant as $h$ approaches $0$ .

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

Step 3.1.2.1.3

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

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

Step 3.1.2.1.4

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

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

Step 3.1.2.1.5

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Combine the opposite terms in $x^{2} - {(x + 0)}^{2}$ .

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Step 3.1.2.3.1

Add $x$ and $0$ .

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

Step 3.1.2.3.2

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

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 3.1.3.5.1

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

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

Step 3.1.3.5.2

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

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

Step 3.1.3.6

Simplify the answer.

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Step 3.1.3.6.1

Add $x$ and $0$ .

$\frac{1}{x^{2}} \cdot \frac{0}{x^{2} \cdot 0}$

Step 3.1.3.6.2

Multiply $x^{2}$ by $0$ .

$\frac{1}{x^{2}} \cdot \frac{0}{0}$

Step 3.1.3.6.3

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

Undefined

$\frac{1}{x^{2}} \cdot \frac{0}{0}$

Step 3.1.3.7

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

Undefined

$\frac{1}{x^{2}} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{x^{2}} \cdot \frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 3.3.3.1

Apply the distributive property.

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

Step 3.3.3.2

Apply the distributive property.

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

Step 3.3.3.3

Apply the distributive property.

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

Step 3.3.4

Simplify and combine like terms.

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Step 3.3.4.1

Simplify each term.

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Step 3.3.4.1.1

Multiply $x$ by $x$ .

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

Step 3.3.4.1.2

Multiply $h$ by $h$ .

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

Step 3.3.4.2

Add $x h$ and $h x$ .

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Step 3.3.4.2.1

Reorder $x$ and $h$ .

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

Step 3.3.4.2.2

Add $h x$ and $h x$ .

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

Step 3.3.5

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

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

Step 3.3.6

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

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

Step 3.3.7

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

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Step 3.3.7.1

Since $- 1$ is constant with respect to $h$ , the derivative of $- (x^{2} + 2 h x + h^{2})$ with respect to $h$ is $- \frac{d}{d h} [x^{2} + 2 h x + h^{2}]$ .

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

Step 3.3.7.2

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

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

Step 3.3.7.3

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

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

Step 3.3.7.4

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

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

Step 3.3.7.5

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

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

Step 3.3.7.6

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

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

Step 3.3.7.7

Multiply $2$ by $1$ .

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

Step 3.3.7.8

Add $0$ and $2 x$ .

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

Step 3.3.8

Simplify.

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Step 3.3.8.1

Apply the distributive property.

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

Step 3.3.8.2

Combine terms.

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Step 3.3.8.2.1

Multiply $2$ by $- 1$ .

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

Step 3.3.8.2.2

Multiply $2$ by $- 1$ .

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

Step 3.3.8.2.3

Subtract $- (- 2 x - 2 h)$ from $0$ .

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

Step 3.3.8.3

Reorder terms.

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

Step 3.3.9

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

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

Step 3.3.10

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 3.3.10.1

Apply the distributive property.

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

Step 3.3.10.2

Apply the distributive property.

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

Step 3.3.10.3

Apply the distributive property.

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

Step 3.3.11

Simplify and combine like terms.

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Step 3.3.11.1

Simplify each term.

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Step 3.3.11.1.1

Multiply $x$ by $x$ .

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

Step 3.3.11.1.2

Multiply $h$ by $h$ .

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

Step 3.3.11.2

Add $x h$ and $h x$ .

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Step 3.3.11.2.1

Reorder $x$ and $h$ .

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

Step 3.3.11.2.2

Add $h x$ and $h x$ .

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

Step 3.3.12

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

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

Step 3.3.13

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

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

Step 3.3.14

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

$\frac{1}{x^{2}} lim_{h \to 0} \frac{- 2 h - 2 x}{x^{2} + 2 h x + h^{2} + h \frac{d}{d h} [x^{2} + 2 h x + h^{2}]}$

Step 3.3.15

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

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

Step 3.3.16

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

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

Step 3.3.17

Add $0$ and $\frac{d}{d h} [2 h x]$ .

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

Step 3.3.18

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

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

Step 3.3.19

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

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

Step 3.3.20

Multiply $2$ by $1$ .

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

Step 3.3.21

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

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

Step 3.3.22

Simplify.

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Step 3.3.22.1

Apply the distributive property.

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

Step 3.3.22.2

Combine terms.

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Step 3.3.22.2.1

Move $2$ to the left of $h$ .

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

Step 3.3.22.2.2

Raise $h$ to the power of $1$ .

$\frac{1}{x^{2}} lim_{h \to 0} \frac{- 2 h - 2 x}{x^{2} + 2 h x + h^{2} + 2 h x + 2 h^{1} h}$

Step 3.3.22.2.3

Raise $h$ to the power of $1$ .

$\frac{1}{x^{2}} lim_{h \to 0} \frac{- 2 h - 2 x}{x^{2} + 2 h x + h^{2} + 2 h x + 2 h^{1} h^{1}}$

Step 3.3.22.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{x^{2}} lim_{h \to 0} \frac{- 2 h - 2 x}{x^{2} + 2 h x + h^{2} + 2 h x + 2 h^{1 + 1}}$

Step 3.3.22.2.5

Add $1$ and $1$ .

$\frac{1}{x^{2}} lim_{h \to 0} \frac{- 2 h - 2 x}{x^{2} + 2 h x + h^{2} + 2 h x + 2 h^{2}}$

Step 3.3.22.2.6

Add $2 h x$ and $2 h x$ .

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

Step 3.3.22.2.7

Add $h^{2}$ and $2 h^{2}$ .

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

Step 3.3.22.3

Reorder terms.

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

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

Step 4.4

Evaluate the limit of $2 x$ which is constant as $h$ approaches $0$ .

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

Step 4.5

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

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

Step 4.6

Move the term $3$ outside of the limit because it is constant with respect to $h$ .

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

Step 4.7

Move the exponent $2$ from $h^{2}$ outside the limit using the Limits Power Rule.

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

Step 4.8

Evaluate the limit of $x^{2}$ which is constant as $h$ approaches $0$ .

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

Step 4.9

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

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

Step 5

Evaluate the limits by plugging in $0$ for all occurrences of $h$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

$\frac{1}{x^{2}} \cdot \frac{- 2 \cdot 0 - 2 x}{3 \cdot 0^{2} + x^{2} + 4 x \cdot 0}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Multiply $- 2$ by $0$ .

$\frac{1}{x^{2}} \cdot \frac{0 - 2 x}{3 \cdot 0^{2} + x^{2} + 4 x \cdot 0}$

Step 6.1.2

Subtract $2 x$ from $0$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{3 \cdot 0^{2} + x^{2} + 4 x \cdot 0}$

Step 6.2

Simplify the denominator.

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Step 6.2.1

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

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{3 \cdot 0 + x^{2} + 4 x \cdot 0}$

Step 6.2.2

Multiply $3$ by $0$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{0 + x^{2} + 4 x \cdot 0}$

Step 6.2.3

Multiply $4 x \cdot 0$ .

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Step 6.2.3.1

Multiply $0$ by $4$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{0 + x^{2} + 0 x}$

Step 6.2.3.2

Multiply $0$ by $x$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{0 + x^{2} + 0}$

Step 6.2.4

Add $0 + x^{2}$ and $0$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{0 + x^{2}}$

Step 6.2.5

Add $0$ and $x^{2}$ .

$\frac{1}{x^{2}} \cdot \frac{- 2 x}{x^{2}}$

Step 6.3

Combine.

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

Step 6.4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 6.4.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.4.2

Add $2$ and $2$ .

$\frac{1 (- 2 x)}{x^{4}}$

Step 6.5

Multiply $- 2 x$ by $1$ .

$\frac{- 2 x}{x^{4}}$

Step 6.6

Cancel the common factor of $x$ and $x^{4}$ .

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Step 6.6.1

Factor $x$ out of $- 2 x$ .

$\frac{x \cdot - 2}{x^{4}}$

Step 6.6.2

Cancel the common factors.

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Step 6.6.2.1

Factor $x$ out of $x^{4}$ .

$\frac{x \cdot - 2}{x \cdot x^{3}}$

Step 6.6.2.2

Cancel the common factor.

$\frac{x \cdot - 2}{x \cdot x^{3}}$

Step 6.6.2.3

Rewrite the expression.

$\frac{- 2}{x^{3}}$

Step 6.7

Move the negative in front of the fraction.

$- \frac{2}{x^{3}}$