Calculus Examples

$\frac{(- p) (- 0.003 {(20 - 0.001 p)}^{- \frac{1}{2}})}{3 {(20 - 0.001 p)}^{\frac{1}{2}}}$

Step 1

Move ${(20 - 0.001 p)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- p \cdot - 0.003}{3 {(20 - 0.001 p)}^{\frac{1}{2}} {(20 - 0.001 p)}^{\frac{1}{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Multiply ${(20 - 0.001 p)}^{\frac{1}{2}}$ by ${(20 - 0.001 p)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1.1

Move ${(20 - 0.001 p)}^{\frac{1}{2}}$ .

$\frac{- p \cdot - 0.003}{3 ({(20 - 0.001 p)}^{\frac{1}{2}} {(20 - 0.001 p)}^{\frac{1}{2}})}$

Step 2.1.2

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

$\frac{- p \cdot - 0.003}{3 {(20 - 0.001 p)}^{\frac{1}{2} + \frac{1}{2}}}$

Step 2.1.3

Combine the numerators over the common denominator.

$\frac{- p \cdot - 0.003}{3 {(20 - 0.001 p)}^{\frac{1 + 1}{2}}}$

Step 2.1.4

Add $1$ and $1$ .

$\frac{- p \cdot - 0.003}{3 {(20 - 0.001 p)}^{\frac{2}{2}}}$

Step 2.1.5

Divide $2$ by $2$ .

$\frac{- p \cdot - 0.003}{3 {(20 - 0.001 p)}^{1}}$

Step 2.2

Simplify $3 {(20 - 0.001 p)}^{1}$ .

$\frac{- p \cdot - 0.003}{3 (20 - 0.001 p)}$

Step 3

Multiply $- 0.003$ by $- 1$ .

$\frac{0.003 p}{3 (20 - 0.001 p)}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $0.001$ out of $20 - 0.001 p$ .

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Step 4.1.1

Factor $0.001$ out of $20$ .

$\frac{0.003 p}{3 (0.001 (20000) - 0.001 p)}$

Step 4.1.2

Factor $0.001$ out of $- 0.001 p$ .

$\frac{0.003 p}{3 (0.001 (20000) + 0.001 (- p))}$

Step 4.1.3

Factor $0.001$ out of $0.001 (20000) + 0.001 (- p)$ .

$\frac{0.003 p}{3 (0.001 (20000 - p))}$

$\frac{0.003 p}{3 \cdot 0.001 (20000 - p)}$

Step 4.2

Multiply $3$ by $0.001$ .

$\frac{0.003 p}{0.003 (20000 - p)}$

Step 5

Cancel the common factor of $0.003$ .

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Step 5.1

Cancel the common factor.

$\frac{0.003 p}{0.003 (20000 - p)}$

Step 5.2

Rewrite the expression.

$\frac{p}{20000 - p}$