Calculus Examples

$h (v) = \sqrt[3]{\frac{27}{p}} {(v + p)}^{\frac{1}{3}} - 3$

Step 1

By the Sum Rule, the derivative of $\sqrt[3]{\frac{27}{p}} {(v + p)}^{\frac{1}{3}} - 3$ with respect to $p$ is $\frac{d}{d p} [\sqrt[3]{\frac{27}{p}} {(v + p)}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$ .

$\frac{d}{d p} [\sqrt[3]{\frac{27}{p}} {(v + p)}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2

Evaluate $\frac{d}{d p} [\sqrt[3]{\frac{27}{p}} {(v + p)}^{\frac{1}{3}}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{27}{p}}$ as ${(\frac{27}{p})}^{\frac{1}{3}}$ .

$\frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}} {(v + p)}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d p} [f (p) g (p)]$ is $f (p) \frac{d}{d p} [g (p)] + g (p) \frac{d}{d p} [f (p)]$ where $f (p) = {(\frac{27}{p})}^{\frac{1}{3}}$ and $g (p) = {(v + p)}^{\frac{1}{3}}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{d}{d p} [{(v + p)}^{\frac{1}{3}}] + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = p^{\frac{1}{3}}$ and $g (p) = v + p$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $v + p$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{3}}] \frac{d}{d p} [v + p]) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = \frac{1}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {u_{1}}^{\frac{1}{3} - 1} \frac{d}{d p} [v + p]) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $v + p$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} \frac{d}{d p} [v + p]) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.4

By the Sum Rule, the derivative of $v + p$ with respect to $p$ is $\frac{d}{d p} [v] + \frac{d}{d p} [p]$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (\frac{d}{d p} [v] + \frac{d}{d p} [p])) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.5

Since $v$ is constant with respect to $p$ , the derivative of $v$ with respect to $p$ is $0$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + \frac{d}{d p} [p])) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.6

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

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} \frac{d}{d p} [{(\frac{27}{p})}^{\frac{1}{3}}] + \frac{d}{d p} [- 3]$

Step 2.7

Differentiate using the chain rule, which states that $\frac{d}{d p} [f (g (p))]$ is $f' (g (p)) g' (p)$ where $f (p) = p^{\frac{1}{3}}$ and $g (p) = \frac{27}{p}$ .

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Step 2.7.1

To apply the Chain Rule, set $u_{2}$ as $\frac{27}{p}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{3}}] \frac{d}{d p} [\frac{27}{p}]) + \frac{d}{d p} [- 3]$

Step 2.7.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{1}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {u_{2}}^{\frac{1}{3} - 1} \frac{d}{d p} [\frac{27}{p}]) + \frac{d}{d p} [- 3]$

Step 2.7.3

Replace all occurrences of $u_{2}$ with $\frac{27}{p}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} \frac{d}{d p} [\frac{27}{p}]) + \frac{d}{d p} [- 3]$

Step 2.8

Since $27$ is constant with respect to $p$ , the derivative of $\frac{27}{p}$ with respect to $p$ is $27 \frac{d}{d p} [\frac{1}{p}]$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 \frac{d}{d p} [\frac{1}{p}])) + \frac{d}{d p} [- 3]$

Step 2.9

Rewrite $\frac{1}{p}$ as $p^{- 1}$ .

Step 2.10

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

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.12

Combine $- 1$ and $\frac{3}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.13

Combine the numerators over the common denominator.

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1 - 1 \cdot 3}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.14

Simplify the numerator.

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Step 2.14.1

Multiply $- 1$ by $3$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{1 - 3}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.14.2

Subtract $3$ from $1$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{\frac{- 2}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.15

Move the negative in front of the fraction.

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{- \frac{2}{3}} (0 + 1)) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.16

Add $0$ and $1$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{1}{3} {(v + p)}^{- \frac{2}{3}} \cdot 1) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.17

Combine $\frac{1}{3}$ and ${(v + p)}^{- \frac{2}{3}}$ .

${(\frac{27}{p})}^{\frac{1}{3}} (\frac{{(v + p)}^{- \frac{2}{3}}}{3} \cdot 1) + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.18

Multiply $\frac{{(v + p)}^{- \frac{2}{3}}}{3}$ by $1$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{{(v + p)}^{- \frac{2}{3}}}{3} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.19

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

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.20

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.21

Combine $- 1$ and $\frac{3}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.22

Combine the numerators over the common denominator.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1 - 1 \cdot 3}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.23

Simplify the numerator.

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Step 2.23.1

Multiply $- 1$ by $3$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{1 - 3}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.23.2

Subtract $3$ from $1$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{\frac{- 2}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.24

Move the negative in front of the fraction.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{- \frac{2}{3}} (27 (- p^{- 2}))) + \frac{d}{d p} [- 3]$

Step 2.25

Multiply $- 1$ by $27$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{1}{3} {(\frac{27}{p})}^{- \frac{2}{3}} (- 27 p^{- 2})) + \frac{d}{d p} [- 3]$

Step 2.26

Combine $- 27$ and $\frac{1}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{- 27}{3} {(\frac{27}{p})}^{- \frac{2}{3}} p^{- 2}) + \frac{d}{d p} [- 3]$

Step 2.27

Combine $p^{- 2}$ and $\frac{- 27}{3}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{p^{- 2} \cdot - 27}{3} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.28

Move $- 27$ to the left of $p^{- 2}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{- 27 \cdot p^{- 2}}{3} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.29

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

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{- 27}{3 p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.30

Cancel the common factor of $- 27$ and $3$ .

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Step 2.30.1

Factor $3$ out of $- 27$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{3 \cdot - 9}{3 p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.30.2

Cancel the common factors.

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Step 2.30.2.1

Factor $3$ out of $3 p^{2}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{3 \cdot - 9}{3 (p^{2})} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.30.2.2

Cancel the common factor.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{3 \cdot - 9}{3 p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.30.2.3

Rewrite the expression.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (\frac{- 9}{p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.31

Move the negative in front of the fraction.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} + {(v + p)}^{\frac{1}{3}} (- \frac{9}{p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}}) + \frac{d}{d p} [- 3]$

Step 2.32

Combine ${(v + p)}^{\frac{1}{3}}$ and $\frac{9}{p^{2}}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}} \cdot 9}{p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}} + \frac{d}{d p} [- 3]$

Step 2.33

Move $9$ to the left of ${(v + p)}^{\frac{1}{3}}$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}} + \frac{d}{d p} [- 3]$

Step 3

Since $- 3$ is constant with respect to $p$ , the derivative of $- 3$ with respect to $p$ is $0$ .

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} {(\frac{27}{p})}^{- \frac{2}{3}} + 0$

Step 4

Simplify.

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Step 4.1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{27}{p})}^{\frac{1}{3}} \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} {(\frac{p}{27})}^{\frac{2}{3}} + 0$

Step 4.2

Apply the product rule to $\frac{27}{p}$ .

$\frac{27^{\frac{1}{3}}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} {(\frac{p}{27})}^{\frac{2}{3}} + 0$

Step 4.3

Apply the product rule to $\frac{p}{27}$ .

$\frac{27^{\frac{1}{3}}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4

Combine terms.

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Step 4.4.1

Rewrite $27$ as $3^{3}$ .

$\frac{{(3^{3})}^{\frac{1}{3}}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.2

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

$\frac{3^{3 (\frac{1}{3})}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.3

Cancel the common factor of $3$ .

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Step 4.4.3.1

Cancel the common factor.

$\frac{3^{3 (\frac{1}{3})}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.3.2

Rewrite the expression.

$\frac{3^{1}}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.4

Evaluate the exponent.

$\frac{3}{p^{\frac{1}{3}}} \cdot \frac{1}{3 {(v + p)}^{\frac{2}{3}}} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.5

Multiply $\frac{3}{p^{\frac{1}{3}}}$ by $\frac{1}{3 {(v + p)}^{\frac{2}{3}}}$ .

$\frac{3}{p^{\frac{1}{3}} (3 {(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.6

Cancel the common factor.

$\frac{3}{p^{\frac{1}{3}} (3 {(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.7

Rewrite the expression.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{27^{\frac{2}{3}}} + 0$

Step 4.4.8

Rewrite $27$ as $3^{3}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{{(3^{3})}^{\frac{2}{3}}} + 0$

Step 4.4.9

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

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{3^{3 (\frac{2}{3})}} + 0$

Step 4.4.10

Cancel the common factor of $3$ .

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Step 4.4.10.1

Cancel the common factor.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{3^{3 (\frac{2}{3})}} + 0$

Step 4.4.10.2

Rewrite the expression.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{3^{2}} + 0$

Step 4.4.11

Raise $3$ to the power of $2$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}} \cdot \frac{p^{\frac{2}{3}}}{9} + 0$

Step 4.4.12

Multiply $\frac{p^{\frac{2}{3}}}{9}$ by $\frac{9 {(v + p)}^{\frac{1}{3}}}{p^{2}}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{p^{\frac{2}{3}} (9 {(v + p)}^{\frac{1}{3}})}{9 p^{2}} + 0$

Step 4.4.13

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

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{2} p^{- \frac{2}{3}}} + 0$

Step 4.4.14

Multiply $p^{2}$ by $p^{- \frac{2}{3}}$ by adding the exponents.

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Step 4.4.14.1

Move $p^{- \frac{2}{3}}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 (p^{- \frac{2}{3}} p^{2})} + 0$

Step 4.4.14.2

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

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{- \frac{2}{3} + 2}} + 0$

Step 4.4.14.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{- \frac{2}{3} + 2 \cdot \frac{3}{3}}} + 0$

Step 4.4.14.4

Combine $2$ and $\frac{3}{3}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{- \frac{2}{3} + \frac{2 \cdot 3}{3}}} + 0$

Step 4.4.14.5

Combine the numerators over the common denominator.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{\frac{- 2 + 2 \cdot 3}{3}}} + 0$

Step 4.4.14.6

Simplify the numerator.

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Step 4.4.14.6.1

Multiply $2$ by $3$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{\frac{- 2 + 6}{3}}} + 0$

Step 4.4.14.6.2

Add $- 2$ and $6$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{\frac{4}{3}}} + 0$

Step 4.4.15

Cancel the common factor.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{9 {(v + p)}^{\frac{1}{3}}}{9 p^{\frac{4}{3}}} + 0$

Step 4.4.16

Rewrite the expression.

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} + 0$

Step 4.4.17

To write $\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})}$ as a fraction with a common denominator, multiply by $\frac{p^{\frac{3}{3}}}{p^{\frac{3}{3}}}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} \cdot \frac{p^{\frac{3}{3}}}{p^{\frac{3}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} + 0$

Step 4.4.18

To write $- \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}}$ .

$\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})} \cdot \frac{p^{\frac{3}{3}}}{p^{\frac{3}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19

Write each expression with a common denominator of $p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.19.1

Multiply $\frac{1}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}})}$ by $\frac{p^{\frac{3}{3}}}{p^{\frac{3}{3}}}$ .

$\frac{p^{\frac{3}{3}}}{p^{\frac{1}{3}} ({(v + p)}^{\frac{2}{3}}) p^{\frac{3}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19.2

Multiply $p^{\frac{1}{3}}$ by $p^{\frac{3}{3}}$ by adding the exponents.

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Step 4.4.19.2.1

Move $p^{\frac{3}{3}}$ .

$\frac{p^{\frac{3}{3}}}{p^{\frac{3}{3}} p^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19.2.2

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

$\frac{p^{\frac{3}{3}}}{p^{\frac{3}{3} + \frac{1}{3}} {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19.2.3

Combine the numerators over the common denominator.

$\frac{p^{\frac{3}{3}}}{p^{\frac{3 + 1}{3}} {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19.2.4

Add $3$ and $1$ .

$\frac{p^{\frac{3}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}} \cdot \frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.19.3

Multiply $\frac{{(v + p)}^{\frac{1}{3}}}{p^{\frac{4}{3}}}$ by $\frac{{(v + p)}^{\frac{2}{3}}}{{(v + p)}^{\frac{2}{3}}}$ .

$\frac{p^{\frac{3}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} - \frac{{(v + p)}^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.20

Combine the numerators over the common denominator.

$\frac{p^{\frac{3}{3}} - {(v + p)}^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.21

Cancel the common factor of $3$ .

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Step 4.4.21.1

Cancel the common factor.

$\frac{p^{\frac{3}{3}} - {(v + p)}^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.21.2

Rewrite the expression.

$\frac{p^{1} - {(v + p)}^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.22

Simplify.

$\frac{p - {(v + p)}^{\frac{1}{3}} {(v + p)}^{\frac{2}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.23

Multiply ${(v + p)}^{\frac{1}{3}}$ by ${(v + p)}^{\frac{2}{3}}$ by adding the exponents.

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Step 4.4.23.1

Move ${(v + p)}^{\frac{2}{3}}$ .

$\frac{p - ({(v + p)}^{\frac{2}{3}} {(v + p)}^{\frac{1}{3}})}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.23.2

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

$\frac{p - {(v + p)}^{\frac{2}{3} + \frac{1}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.23.3

Combine the numerators over the common denominator.

$\frac{p - {(v + p)}^{\frac{2 + 1}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.23.4

Add $2$ and $1$ .

$\frac{p - {(v + p)}^{\frac{3}{3}}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.23.5

Divide $3$ by $3$ .

$\frac{p - {(v + p)}^{1}}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.24

Simplify $- {(v + p)}^{1}$ .

$\frac{p - (v + p)}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}} + 0$

Step 4.4.25

Add $\frac{p - (v + p)}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$ and $0$ .

$\frac{p - (v + p)}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Apply the distributive property.

$\frac{p - v - p}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$

Step 4.5.2

Subtract $p$ from $p$ .

$\frac{- v + 0}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$

Step 4.5.3

Add $- v$ and $0$ .

$\frac{- v}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{v}{p^{\frac{4}{3}} {(v + p)}^{\frac{2}{3}}}$