Calculus Examples

$f (x) = - \frac{\sqrt{y}}{\sqrt{x}}$

Step 1

Multiply $\frac{\sqrt{y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) = - (\frac{\sqrt{y}}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt{y}}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.2

Raise $\sqrt{x}$ to the power of $1$ .

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.3

Raise $\sqrt{x}$ to the power of $1$ .

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{\sqrt{x} \sqrt{x}}$

Step 2.4

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

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{{\sqrt{x}}^{1 + 1}}$

Step 2.5

Add $1$ and $1$ .

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{{\sqrt{x}}^{2}}$

Step 2.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 2.6.1

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

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}$

Step 2.6.2

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

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}$

Step 2.6.3

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

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{x^{\frac{2}{2}}}$

Step 2.6.4.2

Rewrite the expression.

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{x}$

Step 2.6.5

Simplify.

$f (x) = - \frac{\sqrt{y} \sqrt{x}}{x}$

Step 3

Combine using the product rule for radicals.

$f (x) = - \frac{\sqrt{y x}}{x}$

Step 4

Consider the difference quotient formula.

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

Step 5

Find the components of the definition.

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Step 5.1

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

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Step 5.1.1

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

$f (x + h) = - \frac{\sqrt{y (x + h)}}{x + h}$

Step 5.1.2

The final answer is $- \frac{\sqrt{y (x + h)}}{x + h}$ .

$- \frac{\sqrt{y (x + h)}}{x + h}$

Step 5.2

Find the components of the definition.

$f (x + h) = - \frac{\sqrt{y (x + h)}}{x + h}$

$f (x) = - \frac{\sqrt{y x}}{x}$

$f (x + h) = - \frac{\sqrt{y (x + h)}}{x + h}$

$f (x) = - \frac{\sqrt{y x}}{x}$

Step 6

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{\sqrt{y (x + h)}}{x + h} - (- \frac{\sqrt{y x}}{x})}{h}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $- - \frac{\sqrt{y x}}{x}$ .

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Step 7.1.1.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{\sqrt{y (x + h)}}{x + h} + 1 \frac{\sqrt{y x}}{x}}{h}$

Step 7.1.1.2

Multiply $\frac{\sqrt{y x}}{x}$ by $1$ .

$\frac{- \frac{\sqrt{y (x + h)}}{x + h} + \frac{\sqrt{y x}}{x}}{h}$

Step 7.1.2

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

$\frac{- \frac{\sqrt{y (x + h)}}{x + h} \cdot \frac{x}{x} + \frac{\sqrt{y x}}{x}}{h}$

Step 7.1.3

To write $\frac{\sqrt{y x}}{x}$ as a fraction with a common denominator, multiply by $\frac{x + h}{x + h}$ .

$\frac{- \frac{\sqrt{y (x + h)}}{x + h} \cdot \frac{x}{x} + \frac{\sqrt{y x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 7.1.4

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

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Step 7.1.4.1

Multiply $\frac{\sqrt{y (x + h)}}{x + h}$ by $\frac{x}{x}$ .

$\frac{- \frac{\sqrt{y (x + h)} x}{(x + h) x} + \frac{\sqrt{y x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 7.1.4.2

Multiply $\frac{\sqrt{y x}}{x}$ by $\frac{x + h}{x + h}$ .

$\frac{- \frac{\sqrt{y (x + h)} x}{(x + h) x} + \frac{\sqrt{y x} (x + h)}{x (x + h)}}{h}$

Step 7.1.4.3

Reorder the factors of $(x + h) x$ .

$\frac{- \frac{\sqrt{y (x + h)} x}{x (x + h)} + \frac{\sqrt{y x} (x + h)}{x (x + h)}}{h}$

Step 7.1.5

Combine the numerators over the common denominator.

$\frac{\frac{- \sqrt{y (x + h)} x + \sqrt{y x} (x + h)}{x (x + h)}}{h}$

Step 7.1.6

Rewrite $\frac{- \sqrt{y (x + h)} x + \sqrt{y x} (x + h)}{x (x + h)}$ in a factored form.

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Step 7.1.6.1

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

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

Step 7.1.6.2

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

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

Step 7.1.6.3

Apply the distributive property.

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

Step 7.1.6.4

Apply the product rule to $y x$ .

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

Step 7.1.6.5

Apply the distributive property.

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

Step 7.1.6.6

Multiply $x^{\frac{1}{2}}$ by $x$ by adding the exponents.

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Step 7.1.6.6.1

Move $x$ .

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

Step 7.1.6.6.2

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

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Step 7.1.6.6.2.1

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

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

Step 7.1.6.6.2.2

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

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

Step 7.1.6.6.3

Write $1$ as a fraction with a common denominator.

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

Step 7.1.6.6.4

Combine the numerators over the common denominator.

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

Step 7.1.6.6.5

Add $2$ and $1$ .

$\frac{\frac{- {(y x + y h)}^{\frac{1}{2}} x + y^{\frac{1}{2}} x^{\frac{3}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7

Rewrite $- {(y x + y h)}^{\frac{1}{2}} x + y^{\frac{1}{2}} x^{\frac{3}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h$ in a factored form.

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Step 7.1.6.7.1

Factor $x^{\frac{1}{2}}$ out of $- {(y x + y h)}^{\frac{1}{2}} x + y^{\frac{1}{2}} x^{\frac{3}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h$ .

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Step 7.1.6.7.1.1

Reorder the expression.

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Step 7.1.6.7.1.1.1

Reorder $y$ and $x$ .

$\frac{\frac{- {(x y + y h)}^{\frac{1}{2}} x + y^{\frac{1}{2}} x^{\frac{3}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7.1.1.2

Move ${(x y + y h)}^{\frac{1}{2}}$ .

$\frac{\frac{- x {(x y + y h)}^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{3}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7.1.1.3

Reorder $y^{\frac{1}{2}}$ and $x^{\frac{3}{2}}$ .

$\frac{\frac{- x {(x y + y h)}^{\frac{1}{2}} + x^{\frac{3}{2}} y^{\frac{1}{2}} + y^{\frac{1}{2}} x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7.1.1.4

Reorder $y^{\frac{1}{2}}$ and $x^{\frac{1}{2}}$ .

$\frac{\frac{- x {(x y + y h)}^{\frac{1}{2}} + x^{\frac{3}{2}} y^{\frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7.1.2

Factor $x^{\frac{1}{2}}$ out of $- x {(x y + y h)}^{\frac{1}{2}}$ .

$\frac{\frac{x^{\frac{1}{2}} (- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}}) + x^{\frac{3}{2}} y^{\frac{1}{2}} + x^{\frac{1}{2}} y^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 7.1.6.7.1.3

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{3}{2}} y^{\frac{1}{2}}$ .

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

Step 7.1.6.7.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} y^{\frac{1}{2}} h$ .

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

Step 7.1.6.7.1.5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}}) + x^{\frac{1}{2}} (x^{\frac{2}{2}} y^{\frac{1}{2}})$ .

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

Step 7.1.6.7.1.6

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} + x^{\frac{2}{2}} y^{\frac{1}{2}}) + x^{\frac{1}{2}} (y^{\frac{1}{2}} h)$ .

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

Step 7.1.6.7.2

Divide $2$ by $2$ .

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

Step 7.1.6.7.3

Simplify.

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

Step 7.1.6.8

Reduce the expression $\frac{x^{\frac{1}{2}} (- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} + x y^{\frac{1}{2}} + y^{\frac{1}{2}} h)}{x (x + h)}$ by cancelling the common factors.

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Step 7.1.6.8.1

Factor $x$ out of $x^{\frac{1}{2}} (- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} + x y^{\frac{1}{2}} + y^{\frac{1}{2}} h)$ .

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

Step 7.1.6.8.2

Cancel the common factor.

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

Step 7.1.6.8.3

Rewrite the expression.

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

Step 7.1.7

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.3

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

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

Step 7.4

Factor $- 1$ out of $- x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}}$ .

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

Step 7.5

Factor $- 1$ out of $x y^{\frac{1}{2}}$ .

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

Step 7.6

Factor $- 1$ out of $- (x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}}) - (- x y^{\frac{1}{2}})$ .

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

Step 7.7

Factor $- 1$ out of $y^{\frac{1}{2}} h$ .

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

Step 7.8

Factor $- 1$ out of $- (x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} - x y^{\frac{1}{2}}) - (- y^{\frac{1}{2}} h)$ .

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

Step 7.9

Simplify the expression.

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Step 7.9.1

Rewrite $- (x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} - x y^{\frac{1}{2}} - y^{\frac{1}{2}} h)$ as $- 1 (x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} - x y^{\frac{1}{2}} - y^{\frac{1}{2}} h)$ .

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

Step 7.9.2

Move the negative in front of the fraction.

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

Step 7.9.3

Reorder factors in $- \frac{x^{\frac{1}{2}} {(x y + y h)}^{\frac{1}{2}} - x y^{\frac{1}{2}} - y^{\frac{1}{2}} h}{(x + h) x^{\frac{1}{2}} h}$ .

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

Step 8