Calculus Examples

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

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{\sqrt{t + h}}$ by $\frac{\sqrt{t + h}}{\sqrt{t + h}}$ .

$\frac{1}{\sqrt{t + h}} \cdot \frac{\sqrt{t + h}}{\sqrt{t + h}} - (\frac{1}{\sqrt{t}})$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{t + h}}$ by $\frac{\sqrt{t + h}}{\sqrt{t + h}}$ .

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

Step 1.2.2

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

$\frac{\sqrt{t + h}}{{\sqrt{t + h}}^{1} \sqrt{t + h}} - (\frac{1}{\sqrt{t}})$

Step 1.2.3

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

$\frac{\sqrt{t + h}}{{\sqrt{t + h}}^{1} {\sqrt{t + h}}^{1}} - (\frac{1}{\sqrt{t}})$

Step 1.2.4

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

$\frac{\sqrt{t + h}}{{\sqrt{t + h}}^{1 + 1}} - (\frac{1}{\sqrt{t}})$

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

$\frac{\sqrt{t + h}}{{(t + h)}^{\frac{1}{2} \cdot 2}} - (\frac{1}{\sqrt{t}})$

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{t + h}}{{(t + h)}^{1}} - (\frac{1}{\sqrt{t}})$

Step 1.2.6.5

Simplify.

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

Step 1.3

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

$\frac{\sqrt{t + h}}{t + h} - (\frac{1}{\sqrt{t}} \cdot \frac{\sqrt{t}}{\sqrt{t}})$

Step 1.4

Combine and simplify the denominator.

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Step 1.4.1

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{\sqrt{t} \sqrt{t}}$

Step 1.4.2

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{{\sqrt{t}}^{1} \sqrt{t}}$

Step 1.4.3

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{{\sqrt{t}}^{1} {\sqrt{t}}^{1}}$

Step 1.4.4

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{{\sqrt{t}}^{1 + 1}}$

Step 1.4.5

Add $1$ and $1$ .

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{{\sqrt{t}}^{2}}$

Step 1.4.6

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

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Step 1.4.6.1

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

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

Step 1.4.6.2

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{t^{\frac{1}{2} \cdot 2}}$

Step 1.4.6.3

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

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{t^{\frac{2}{2}}}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{t^{\frac{2}{2}}}$

Step 1.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{t^{1}}$

Step 1.4.6.5

Simplify.

$\frac{\sqrt{t + h}}{t + h} - \frac{\sqrt{t}}{t}$

Step 2

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

$\frac{\sqrt{t + h}}{t + h} \cdot \frac{t}{t} - \frac{\sqrt{t}}{t}$

Step 3

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

$\frac{\sqrt{t + h}}{t + h} \cdot \frac{t}{t} - \frac{\sqrt{t}}{t} \cdot \frac{t + h}{t + h}$

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$\frac{\sqrt{t + h} t}{(t + h) t} - \frac{\sqrt{t} (t + h)}{t (t + h)}$

Step 4.3

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

$\frac{\sqrt{t + h} t}{t (t + h)} - \frac{\sqrt{t} (t + h)}{t (t + h)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt{t + h} t - \sqrt{t} (t + h)}{t (t + h)}$

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Apply the distributive property.

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

Step 6.4

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

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Step 6.4.1

Move $t$ .

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

Step 6.4.2

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

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Step 6.4.2.1

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

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

Step 6.4.2.2

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

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

Step 6.4.3

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

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

Step 6.4.4

Combine the numerators over the common denominator.

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

Step 6.4.5

Add $2$ and $1$ .

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

Step 6.5

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

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Step 6.5.1

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

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Step 6.5.1.1

Reorder the expression.

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Step 6.5.1.1.1

Reorder $t$ and $h$ .

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

Step 6.5.1.1.2

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

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

Step 6.5.1.1.3

Move $t^{\frac{1}{2}}$ .

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

Step 6.5.1.2

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

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

Step 6.5.1.3

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

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

Step 6.5.1.4

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

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

Step 6.5.1.5

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

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

Step 6.5.1.6

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

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

Step 6.5.2

Divide $2$ by $2$ .

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

Step 6.5.3

Simplify.

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

Step 7

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

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

Step 8

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

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Step 8.1

Move $t^{- \frac{1}{2}}$ .

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

Step 8.2

Multiply $t^{- \frac{1}{2}}$ by $t$ .

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Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

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

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 8.5

Add $- 1$ and $2$ .

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