Calculus Examples

$lim_{h \to 0} \frac{5 {(x + h)}^{3} - (5 x^{3})}{h}$

Step 1

Multiply $5$ by $- 1$ .

$lim_{h \to 0} \frac{5 {(x + h)}^{3} - 5 x^{3}}{h}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{h \to 0} 5 {(x + h)}^{3} - 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

Evaluate the limit.

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Step 2.1.2.1.1

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

$\frac{lim_{h \to 0} 5 {(x + h)}^{3} - lim_{h \to 0} 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.1.2

Move the term $5$ outside of the limit because it is constant with respect to $h$ .

$\frac{5 lim_{h \to 0} {(x + h)}^{3} - lim_{h \to 0} 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.1.3

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

$\frac{5 {(lim_{h \to 0} x + h)}^{3} - lim_{h \to 0} 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.1.4

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

$\frac{5 {(lim_{h \to 0} x + lim_{h \to 0} h)}^{3} - lim_{h \to 0} 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.1.5

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

$\frac{5 {(x + lim_{h \to 0} h)}^{3} - lim_{h \to 0} 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.1.6

Evaluate the limit of $5 x^{3}$ which is constant as $h$ approaches $0$ .

$\frac{5 {(x + lim_{h \to 0} h)}^{3} - 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.2

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

$\frac{5 {(x + 0)}^{3} - 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.3

Combine the opposite terms in $5 {(x + 0)}^{3} - 5 x^{3}$ .

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Step 2.1.2.3.1

Add $x$ and $0$ .

$\frac{5 x^{3} - 5 x^{3}}{lim_{h \to 0} h}$

Step 2.1.2.3.2

Subtract $5 x^{3}$ from $5 x^{3}$ .

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

Step 2.1.3

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

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{h \to 0} \frac{5 {(x + h)}^{3} - 5 x^{3}}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [5 {(x + h)}^{3} - 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [5 {(x + h)}^{3} - 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.2

By the Sum Rule, the derivative of $5 {(x + h)}^{3} - 5 x^{3}$ with respect to $h$ is $\frac{d}{d h} [5 {(x + h)}^{3}] + \frac{d}{d h} [- 5 x^{3}]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [5 {(x + h)}^{3}] + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3

Evaluate $\frac{d}{d h} [5 {(x + h)}^{3}]$ .

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Step 2.3.3.1

Since $5$ is constant with respect to $h$ , the derivative of $5 {(x + h)}^{3}$ with respect to $h$ is $5 \frac{d}{d h} [{(x + h)}^{3}]$ .

$lim_{h \to 0} \frac{5 \frac{d}{d h} [{(x + h)}^{3}] + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.2

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{3}$ and $g (h) = x + h$ .

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Step 2.3.3.2.1

To apply the Chain Rule, set $u$ as $x + h$ .

$lim_{h \to 0} \frac{5 (\frac{d}{d u} [u^{3}] \frac{d}{d h} [x + h]) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$lim_{h \to 0} \frac{5 (3 u^{2} \frac{d}{d h} [x + h]) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.2.3

Replace all occurrences of $u$ with $x + h$ .

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2} \frac{d}{d h} [x + h]) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.3

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

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2} (\frac{d}{d h} [x] + \frac{d}{d h} [h])) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.4

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

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2} (0 + \frac{d}{d h} [h])) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.5

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

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2} (0 + 1)) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.6

Add $0$ and $1$ .

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2} \cdot 1) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.7

Multiply $3$ by $1$ .

$lim_{h \to 0} \frac{5 (3 {(x + h)}^{2}) + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.3.8

Multiply $3$ by $5$ .

$lim_{h \to 0} \frac{15 {(x + h)}^{2} + \frac{d}{d h} [- 5 x^{3}]}{\frac{d}{d h} [h]}$

Step 2.3.4

Since $- 5 x^{3}$ is constant with respect to $h$ , the derivative of $- 5 x^{3}$ with respect to $h$ is $0$ .

$lim_{h \to 0} \frac{15 {(x + h)}^{2} + 0}{\frac{d}{d h} [h]}$

Step 2.3.5

Add $15 {(x + h)}^{2}$ and $0$ .

$lim_{h \to 0} \frac{15 {(x + h)}^{2}}{\frac{d}{d h} [h]}$

Step 2.3.6

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

$lim_{h \to 0} \frac{15 {(x + h)}^{2}}{1}$

Step 2.4

Divide $15 {(x + h)}^{2}$ by $1$ .

$lim_{h \to 0} 15 {(x + h)}^{2}$

Step 3

Evaluate the limit.

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Step 3.1

Move the term $15$ outside of the limit because it is constant with respect to $h$ .

$15 lim_{h \to 0} {(x + h)}^{2}$

Step 3.2

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

$15 {(lim_{h \to 0} x + h)}^{2}$

Step 3.3

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

$15 {(lim_{h \to 0} x + lim_{h \to 0} h)}^{2}$

Step 3.4

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

$15 {(x + lim_{h \to 0} h)}^{2}$

Step 4

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

$15 {(x + 0)}^{2}$

Step 5

Add $x$ and $0$ .

$15 x^{2}$