Calculus Examples

$H (u) = (u - \sqrt{u}) (u + \sqrt{u})$

Step 1

Consider the difference quotient formula.

$\frac{f (u + h) - f u}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = u + h$ .

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Step 2.1.1

Replace the variable $u$ with $u + h$ in the expression.

$H (u + h) = ((u + h) - \sqrt{u + h}) ((u + h) + \sqrt{u + h})$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Expand $(u + h - \sqrt{u + h}) (u + h + \sqrt{u + h})$ by multiplying each term in the first expression by each term in the second expression.

$H (u + h) = u \cdot u + u h + u \sqrt{u + h} + h u + h \cdot h + h \sqrt{u + h} - \sqrt{u + h} u - \sqrt{u + h} h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2

Simplify terms.

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Step 2.1.2.2.1

Combine the opposite terms in $u \cdot u + u h + u \sqrt{u + h} + h u + h \cdot h + h \sqrt{u + h} - \sqrt{u + h} u - \sqrt{u + h} h - \sqrt{u + h} \sqrt{u + h}$ .

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Step 2.1.2.2.1.1

Reorder the factors in the terms $u \sqrt{u + h}$ and $- \sqrt{u + h} u$ .

$H (u + h) = u \cdot u + u h + u \sqrt{u + h} + h u + h \cdot h + h \sqrt{u + h} - u \sqrt{u + h} - \sqrt{u + h} h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.1.2

Subtract $u \sqrt{u + h}$ from $u \sqrt{u + h}$ .

$H (u + h) = u \cdot u + u h + h u + h \cdot h + h \sqrt{u + h} + 0 - \sqrt{u + h} h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.1.3

Add $u \cdot u + u h + h u + h \cdot h + h \sqrt{u + h}$ and $0$ .

$H (u + h) = u \cdot u + u h + h u + h \cdot h + h \sqrt{u + h} - \sqrt{u + h} h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.1.4

Reorder the factors in the terms $h \sqrt{u + h}$ and $- \sqrt{u + h} h$ .

$H (u + h) = u \cdot u + u h + h u + h \cdot h + h \sqrt{u + h} - h \sqrt{u + h} - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.1.5

Subtract $h \sqrt{u + h}$ from $h \sqrt{u + h}$ .

$H (u + h) = u \cdot u + u h + h u + h \cdot h + 0 - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.1.6

Add $u \cdot u + u h + h u + h \cdot h$ and $0$ .

$H (u + h) = u \cdot u + u h + h u + h \cdot h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.2

Simplify each term.

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Step 2.1.2.2.2.1

Multiply $u$ by $u$ .

$H (u + h) = u^{2} + u h + h u + h \cdot h - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.2.2

Multiply $h$ by $h$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - \sqrt{u + h} \sqrt{u + h}$

Step 2.1.2.2.2.3

Multiply $- \sqrt{u + h} \sqrt{u + h}$ .

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Step 2.1.2.2.2.3.1

Raise $\sqrt{u + h}$ to the power of $1$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - (\sqrt{u + h} \sqrt{u + h})$

Step 2.1.2.2.2.3.2

Raise $\sqrt{u + h}$ to the power of $1$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - (\sqrt{u + h} \sqrt{u + h})$

Step 2.1.2.2.2.3.3

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

$H (u + h) = u^{2} + u h + h u + h^{2} - {\sqrt{u + h}}^{1 + 1}$

Step 2.1.2.2.2.3.4

Add $1$ and $1$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - {\sqrt{u + h}}^{2}$

Step 2.1.2.2.2.4

Rewrite ${\sqrt{u + h}}^{2}$ as $u + h$ .

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Step 2.1.2.2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u + h}$ as ${(u + h)}^{\frac{1}{2}}$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - {({(u + h)}^{\frac{1}{2}})}^{2}$

Step 2.1.2.2.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - {(u + h)}^{\frac{1}{2} \cdot 2}$

Step 2.1.2.2.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$H (u + h) = u^{2} + u h + h u + h^{2} - {(u + h)}^{\frac{2}{2}}$

Step 2.1.2.2.2.4.4

Cancel the common factor of $2$ .

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Step 2.1.2.2.2.4.4.1

Cancel the common factor.

$H (u + h) = u^{2} + u h + h u + h^{2} - {(u + h)}^{\frac{2}{2}}$

Step 2.1.2.2.2.4.4.2

Rewrite the expression.

$H (u + h) = u^{2} + u h + h u + h^{2} - (u + h)$

Step 2.1.2.2.2.4.5

Simplify.

$H (u + h) = u^{2} + u h + h u + h^{2} - (u + h)$

Step 2.1.2.2.2.5

Apply the distributive property.

$H (u + h) = u^{2} + u h + h u + h^{2} - u - h$

Step 2.1.2.3

Add $u h$ and $h u$ .

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Step 2.1.2.3.1

Reorder $u$ and $h$ .

$H (u + h) = u^{2} + h u + h u + h^{2} - u - h$

Step 2.1.2.3.2

Add $h u$ and $h u$ .

$H (u + h) = u^{2} + 2 h u + h^{2} - u - h$

Step 2.1.2.4

The final answer is $u^{2} + 2 h u + h^{2} - u - h$ .

$u^{2} + 2 h u + h^{2} - u - h$

Step 2.2

Reorder.

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Step 2.2.1

Move $- u$ .

$u^{2} + 2 h u + h^{2} - h - u$

Step 2.2.2

Move $u^{2}$ .

$2 h u + h^{2} + u^{2} - h - u$

Step 2.2.3

Reorder $2 h u$ and $h^{2}$ .

$h^{2} + 2 h u + u^{2} - h - u$

Step 2.3

Find the components of the definition.

$H (u + h) = h^{2} + 2 h u + u^{2} - h - u$

$H (u) = u^{2} - u$

$H (u + h) = h^{2} + 2 h u + u^{2} - h - u$

$H (u) = u^{2} - u$

Step 3

Plug in the components.

$\frac{H (u + h) - H u}{h} = \frac{h^{2} + 2 h u + u^{2} - h - u - (u^{2} - u)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{h^{2} + 2 h u + u^{2} - h - u - u^{2} - - u}{h}$

Step 4.1.2

Multiply $- - u$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{h^{2} + 2 h u + u^{2} - h - u - u^{2} + 1 u}{h}$

Step 4.1.2.2

Multiply $u$ by $1$ .

$\frac{h^{2} + 2 h u + u^{2} - h - u - u^{2} + u}{h}$

Step 4.1.3

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

$\frac{h^{2} + 2 h u - h - u + 0 + u}{h}$

Step 4.1.4

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

$\frac{h^{2} + 2 h u - h - u + u}{h}$

Step 4.1.5

Add $- u$ and $u$ .

$\frac{h^{2} + 2 h u - h + 0}{h}$

Step 4.1.6

Add $h^{2} + 2 h u - h$ and $0$ .

$\frac{h^{2} + 2 h u - h}{h}$

Step 4.1.7

Factor $h$ out of $h^{2} + 2 h u - h$ .

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Step 4.1.7.1

Factor $h$ out of $h^{2}$ .

$\frac{h \cdot h + 2 h u - h}{h}$

Step 4.1.7.2

Factor $h$ out of $2 h u$ .

$\frac{h (h) + h (2 u) - h}{h}$

Step 4.1.7.3

Factor $h$ out of $- h$ .

$\frac{h (h) + h (2 u) + h \cdot - 1}{h}$

Step 4.1.7.4

Factor $h$ out of $h (h) + h (2 u)$ .

$\frac{h (h + 2 u) + h \cdot - 1}{h}$

Step 4.1.7.5

Factor $h$ out of $h (h + 2 u) + h \cdot - 1$ .

$\frac{h (h + 2 u - 1)}{h}$

Step 4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.1

Cancel the common factor of $h$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{h (h + 2 u - 1)}{h}$

Step 4.2.1.2

Divide $h + 2 u - 1$ by $1$ .

$h + 2 u - 1$

Step 4.2.2

Reorder $h$ and $2 u$ .

$2 u + h - 1$

Step 5