Calculus Examples

$f (x) = \frac{1}{\sqrt{x}}$

Step 1

Consider the limit definition of the derivative.

$f' (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Step 2

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

$f (x) = \frac{1}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Find the components of the definition.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

The final answer is $\frac{\sqrt{x + h}}{x + h}$ .

$\frac{\sqrt{x + h}}{x + h}$

Step 4.2

Find the components of the definition.

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

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

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

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

Step 5

Plug in the components.

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h}}{x + h} - (\frac{\sqrt{x}}{x})}{h}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h}}{x + h} \cdot \frac{x}{x} - \frac{\sqrt{x}}{x}}{h}$

Step 6.1.2

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

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h}}{x + h} \cdot \frac{x}{x} - \frac{\sqrt{x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 6.1.3

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 6.1.3.1

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

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h} x}{(x + h) x} - \frac{\sqrt{x}}{x} \cdot \frac{x + h}{x + h}}{h}$

Step 6.1.3.2

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

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h} x}{(x + h) x} - \frac{\sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.3.3

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

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h} x}{x (x + h)} - \frac{\sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.4

Combine the numerators over the common denominator.

$f' (x) = lim_{h \to 0} \frac{\frac{\sqrt{x + h} x - \sqrt{x} (x + h)}{x (x + h)}}{h}$

Step 6.1.5

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

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Step 6.1.5.1

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

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

Step 6.1.5.2

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

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

Step 6.1.5.3

Apply the distributive property.

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

Step 6.1.5.4

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

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Step 6.1.5.4.1

Move $x$ .

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

Step 6.1.5.4.2

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

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Step 6.1.5.4.2.1

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

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

Step 6.1.5.4.2.2

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

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

Step 6.1.5.4.3

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

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

Step 6.1.5.4.4

Combine the numerators over the common denominator.

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

Step 6.1.5.4.5

Add $2$ and $1$ .

$f' (x) = lim_{h \to 0} \frac{\frac{{(x + h)}^{\frac{1}{2}} x - x^{\frac{3}{2}} - x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 6.1.5.5

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

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Step 6.1.5.5.1

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

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Step 6.1.5.5.1.1

Reorder ${(x + h)}^{\frac{1}{2}}$ and $x$ .

$f' (x) = lim_{h \to 0} \frac{\frac{x {(x + h)}^{\frac{1}{2}} - x^{\frac{3}{2}} - x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 6.1.5.5.1.2

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

$f' (x) = lim_{h \to 0} \frac{\frac{x^{\frac{1}{2}} (x^{\frac{1}{2}} {(x + h)}^{\frac{1}{2}}) - x^{\frac{3}{2}} - x^{\frac{1}{2}} h}{x (x + h)}}{h}$

Step 6.1.5.5.1.3

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

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

Step 6.1.5.5.1.4

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

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

Step 6.1.5.5.1.5

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

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

Step 6.1.5.5.1.6

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

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

Step 6.1.5.5.2

Divide $2$ by $2$ .

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

Step 6.1.5.5.3

Simplify.

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

Step 6.1.5.6

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

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Step 6.1.5.6.1

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

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

Step 6.1.5.6.2

Cancel the common factor.

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

Step 6.1.5.6.3

Rewrite the expression.

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

Step 6.1.6

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

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3

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

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

Step 6.4

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

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

Step 7

Simplify the limit argument.

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Step 7.1

Convert fractional exponents to radicals.

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Step 7.1.1

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 7.1.2

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

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

Step 7.1.3

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 7.2

Combine using the product rule for radicals.

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

Step 8

Since the numerator is negative and the denominator $h \sqrt{x} (x + h)$ approaches zero and is less than zero for $h$ near $0$ on both sides, the function increases without bound.

$\infty$

Step 9