Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $\sqrt{t} - \frac{1}{\sqrt{t}}$ with respect to $t$ is $\frac{d}{d t} [\sqrt{t}] + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$ .

$\frac{d}{d t} [\sqrt{t}] + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2

Evaluate $\frac{d}{d t} [\sqrt{t}]$ .

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Step 2.1

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

$\frac{d}{d t} [t^{\frac{1}{2}}] + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} t^{\frac{1}{2} - 1} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.4

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

$\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.5

Combine the numerators over the common denominator.

$\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} t^{\frac{1 - 2}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} t^{\frac{- 1}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 2.7

Move the negative in front of the fraction.

$\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [- \frac{1}{\sqrt{t}}]$

Step 3

Evaluate $\frac{d}{d t} [- \frac{1}{\sqrt{t}}]$ .

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Step 3.1

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

$\frac{1}{2} t^{- \frac{1}{2}} + \frac{d}{d t} [- \frac{1}{t^{\frac{1}{2}}}]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = - 1$ and $g (t) = \frac{1}{t^{\frac{1}{2}}}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - \frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}}] + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.3

Rewrite $\frac{1}{t^{\frac{1}{2}}}$ as ${(t^{\frac{1}{2}})}^{- 1}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - \frac{d}{d t} [{(t^{\frac{1}{2}})}^{- 1}] + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = t^{\frac{1}{2}}$ .

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Step 3.4.1

To apply the Chain Rule, set $u$ as $t^{\frac{1}{2}}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{\frac{1}{2}}]) + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.4.2

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

$\frac{1}{2} t^{- \frac{1}{2}} - (- u^{- 2} \frac{d}{d t} [t^{\frac{1}{2}}]) + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.4.3

Replace all occurrences of $u$ with $t^{\frac{1}{2}}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- {(t^{\frac{1}{2}})}^{- 2} \frac{d}{d t} [t^{\frac{1}{2}}]) + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.5

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- {(t^{\frac{1}{2}})}^{- 2} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \frac{d}{d t} [- 1]$

Step 3.6

Since $- 1$ is constant with respect to $t$ , the derivative of $- 1$ with respect to $t$ is $0$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- {(t^{\frac{1}{2}})}^{- 2} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.7

Multiply the exponents in ${(t^{\frac{1}{2}})}^{- 2}$ .

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Step 3.7.1

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

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{\frac{1}{2} \cdot - 2} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.7.2

Cancel the common factor of $2$ .

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Step 3.7.2.1

Factor $2$ out of $- 2$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.7.2.2

Cancel the common factor.

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.7.2.3

Rewrite the expression.

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} - 1})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.8

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.9

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

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.10

Combine the numerators over the common denominator.

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{1 - 2}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.11.2

Subtract $2$ from $1$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{\frac{- 1}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.12

Move the negative in front of the fraction.

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} (\frac{1}{2} t^{- \frac{1}{2}})) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.13

Combine $\frac{1}{2}$ and $t^{- \frac{1}{2}}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - (- t^{- 1} \frac{t^{- \frac{1}{2}}}{2}) + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.14

Combine $\frac{t^{- \frac{1}{2}}}{2}$ and $t^{- 1}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{- \frac{1}{2}} t^{- 1}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15

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

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Step 3.15.1

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

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{- \frac{1}{2} - 1}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.3

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

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.4

Combine the numerators over the common denominator.

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{\frac{- 1 - 1 \cdot 2}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.5

Simplify the numerator.

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Step 3.15.5.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{\frac{- 1 - 2}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.5.2

Subtract $2$ from $- 1$ .

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{\frac{- 3}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.15.6

Move the negative in front of the fraction.

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{t^{- \frac{3}{2}}}{2} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.16

Move $t^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} t^{- \frac{1}{2}} - - \frac{1}{2 t^{\frac{3}{2}}} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.17

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} t^{- \frac{1}{2}} + 1 \frac{1}{2 t^{\frac{3}{2}}} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.18

Multiply $\frac{1}{2 t^{\frac{3}{2}}}$ by $1$ .

$\frac{1}{2} t^{- \frac{1}{2}} + \frac{1}{2 t^{\frac{3}{2}}} + \frac{1}{t^{\frac{1}{2}}} \cdot 0$

Step 3.19

Multiply $\frac{1}{t^{\frac{1}{2}}}$ by $0$ .

$\frac{1}{2} t^{- \frac{1}{2}} + \frac{1}{2 t^{\frac{3}{2}}} + 0$

Step 3.20

Add $\frac{1}{2 t^{\frac{3}{2}}}$ and $0$ .

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

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{t^{\frac{1}{2}}} + \frac{1}{2 t^{\frac{3}{2}}}$

Step 4.2

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

$\frac{1}{2 t^{\frac{1}{2}}} + \frac{1}{2 t^{\frac{3}{2}}}$