Calculus Examples

$f (x) = 50 \frac{1}{2} \cdot p^{- \frac{1}{2}}$

Step 1

Convert $50 \frac{1}{2}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$\frac{d}{d p} [(50 + \frac{1}{2}) \cdot p^{- \frac{1}{2}}]$

Step 1.2

Add $50$ and $\frac{1}{2}$ .

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Step 1.2.1

To write $50$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d p} [(50 \cdot \frac{2}{2} + \frac{1}{2}) \cdot p^{- \frac{1}{2}}]$

Step 1.2.2

Combine $50$ and $\frac{2}{2}$ .

$\frac{d}{d p} [(\frac{50 \cdot 2}{2} + \frac{1}{2}) \cdot p^{- \frac{1}{2}}]$

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{d}{d p} [\frac{50 \cdot 2 + 1}{2} \cdot p^{- \frac{1}{2}}]$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $50$ by $2$ .

$\frac{d}{d p} [\frac{100 + 1}{2} \cdot p^{- \frac{1}{2}}]$

Step 1.2.4.2

Add $100$ and $1$ .

$\frac{d}{d p} [\frac{101}{2} \cdot p^{- \frac{1}{2}}]$

Step 2

Since $\frac{101}{2}$ is constant with respect to $p$ , the derivative of $\frac{101}{2} \cdot p^{- \frac{1}{2}}$ with respect to $p$ is $\frac{101}{2} \frac{d}{d p} [p^{- \frac{1}{2}}]$ .

$\frac{101}{2} \frac{d}{d p} [p^{- \frac{1}{2}}]$

Step 3

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

$\frac{101}{2} (- \frac{1}{2} p^{- \frac{1}{2} - 1})$

Step 4

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

$\frac{101}{2} (- \frac{1}{2} p^{- \frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 5

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

$\frac{101}{2} (- \frac{1}{2} p^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 6

Combine the numerators over the common denominator.

$\frac{101}{2} (- \frac{1}{2} p^{\frac{- 1 - 1 \cdot 2}{2}})$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$\frac{101}{2} (- \frac{1}{2} p^{\frac{- 1 - 2}{2}})$

Step 7.2

Subtract $2$ from $- 1$ .

$\frac{101}{2} (- \frac{1}{2} p^{\frac{- 3}{2}})$

Step 8

Move the negative in front of the fraction.

$\frac{101}{2} (- \frac{1}{2} p^{- \frac{3}{2}})$

Step 9

Combine $p^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$\frac{101}{2} (- \frac{p^{- \frac{3}{2}}}{2})$

Step 10

Multiply $\frac{101}{2}$ by $\frac{p^{- \frac{3}{2}}}{2}$ .

$- \frac{101 p^{- \frac{3}{2}}}{2 \cdot 2}$

Step 11

Multiply.

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Step 11.1

Multiply $2$ by $2$ .

$- \frac{101 p^{- \frac{3}{2}}}{4}$

Step 11.2

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

$- \frac{101}{4 p^{\frac{3}{2}}}$