Calculus Examples

$860 - \frac{326}{\sqrt{t}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $860 - \frac{326}{\sqrt{t}}$ with respect to $t$ is $\frac{d}{d t} [860] + \frac{d}{d t} [- \frac{326}{\sqrt{t}}]$ .

$\frac{d}{d t} [860] + \frac{d}{d t} [- \frac{326}{\sqrt{t}}]$

Step 1.2

Since $860$ is constant with respect to $t$ , the derivative of $860$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [- \frac{326}{\sqrt{t}}]$

Step 2

Evaluate $\frac{d}{d t} [- \frac{326}{\sqrt{t}}]$ .

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Step 2.1

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

$0 + \frac{d}{d t} [- \frac{326}{t^{\frac{1}{2}}}]$

Step 2.2

Since $- 326$ is constant with respect to $t$ , the derivative of $- \frac{326}{t^{\frac{1}{2}}}$ with respect to $t$ is $- 326 \frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}}]$ .

$0 - 326 \frac{d}{d t} [\frac{1}{t^{\frac{1}{2}}}]$

Step 2.3

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

$0 - 326 \frac{d}{d t} [{(t^{\frac{1}{2}})}^{- 1}]$

Step 2.4

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 2.4.1

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

$0 - 326 (\frac{d}{d u} [u^{- 1}] \frac{d}{d t} [t^{\frac{1}{2}}])$

Step 2.4.2

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

$0 - 326 (- u^{- 2} \frac{d}{d t} [t^{\frac{1}{2}}])$

Step 2.4.3

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

$0 - 326 (- {(t^{\frac{1}{2}})}^{- 2} \frac{d}{d t} [t^{\frac{1}{2}}])$

Step 2.5

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

$0 - 326 (- {(t^{\frac{1}{2}})}^{- 2} (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 2.6

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

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Step 2.6.1

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

$0 - 326 (- t^{\frac{1}{2} \cdot - 2} (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 2.6.2

Cancel the common factor of $2$ .

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Step 2.6.2.1

Factor $2$ out of $- 2$ .

$0 - 326 (- t^{\frac{1}{2} \cdot (2 (- 1))} (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 2.6.2.2

Cancel the common factor.

$0 - 326 (- t^{\frac{1}{2} \cdot (2 \cdot - 1)} (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 2.6.2.3

Rewrite the expression.

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} - 1}))$

Step 2.7

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

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))$

Step 2.8

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

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}))$

Step 2.9

Combine the numerators over the common denominator.

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{1 - 1 \cdot 2}{2}}))$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Multiply $- 1$ by $2$ .

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{1 - 2}{2}}))$

Step 2.10.2

Subtract $2$ from $1$ .

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{\frac{- 1}{2}}))$

Step 2.11

Move the negative in front of the fraction.

$0 - 326 (- t^{- 1} (\frac{1}{2} t^{- \frac{1}{2}}))$

Step 2.12

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

$0 - 326 (- t^{- 1} \frac{t^{- \frac{1}{2}}}{2})$

Step 2.13

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

$0 - 326 (- \frac{t^{- \frac{1}{2}} t^{- 1}}{2})$

Step 2.14

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

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Step 2.14.1

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

$0 - 326 (- \frac{t^{- \frac{1}{2} - 1}}{2})$

Step 2.14.2

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

$0 - 326 (- \frac{t^{- \frac{1}{2} - 1 \cdot \frac{2}{2}}}{2})$

Step 2.14.3

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

$0 - 326 (- \frac{t^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}}}{2})$

Step 2.14.4

Combine the numerators over the common denominator.

$0 - 326 (- \frac{t^{\frac{- 1 - 1 \cdot 2}{2}}}{2})$

Step 2.14.5

Simplify the numerator.

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Step 2.14.5.1

Multiply $- 1$ by $2$ .

$0 - 326 (- \frac{t^{\frac{- 1 - 2}{2}}}{2})$

Step 2.14.5.2

Subtract $2$ from $- 1$ .

$0 - 326 (- \frac{t^{\frac{- 3}{2}}}{2})$

Step 2.14.6

Move the negative in front of the fraction.

$0 - 326 (- \frac{t^{- \frac{3}{2}}}{2})$

Step 2.15

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

$0 - 326 (- \frac{1}{2 t^{\frac{3}{2}}})$

Step 2.16

Multiply $- 1$ by $- 326$ .

$0 + 326 \frac{1}{2 t^{\frac{3}{2}}}$

Step 2.17

Combine $326$ and $\frac{1}{2 t^{\frac{3}{2}}}$ .

$0 + \frac{326}{2 t^{\frac{3}{2}}}$

Step 2.18

Factor $2$ out of $326$ .

$0 + \frac{2 \cdot 163}{2 t^{\frac{3}{2}}}$

Step 2.19

Cancel the common factors.

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Step 2.19.1

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

$0 + \frac{2 \cdot 163}{2 (t^{\frac{3}{2}})}$

Step 2.19.2

Cancel the common factor.

$0 + \frac{2 \cdot 163}{2 t^{\frac{3}{2}}}$

Step 2.19.3

Rewrite the expression.

$0 + \frac{163}{t^{\frac{3}{2}}}$

Step 3

Add $0$ and $\frac{163}{t^{\frac{3}{2}}}$ .

$\frac{163}{t^{\frac{3}{2}}}$