Calculus Examples

$x = \sqrt[3]{15000 - p^{3}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{15000 - p^{3}}$ as ${(15000 - p^{3})}^{\frac{1}{3}}$ .

$x = {(15000 - p^{3})}^{\frac{1}{3}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (x) = \frac{d}{d x} ({(15000 - p^{3})}^{\frac{1}{3}})$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{3}}$ and $g (x) = 15000 - p^{3}$ .

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Step 4.1.1

To apply the Chain Rule, set $u_{1}$ as $15000 - p^{3}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{3}}] \frac{d}{d x} [15000 - p^{3}]$

Step 4.1.2

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

$\frac{1}{3} {u_{1}}^{\frac{1}{3} - 1} \frac{d}{d x} [15000 - p^{3}]$

Step 4.1.3

Replace all occurrences of $u_{1}$ with $15000 - p^{3}$ .

$\frac{1}{3} {(15000 - p^{3})}^{\frac{1}{3} - 1} \frac{d}{d x} [15000 - p^{3}]$

Step 4.2

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

$\frac{1}{3} {(15000 - p^{3})}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.3

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

$\frac{1}{3} {(15000 - p^{3})}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{1}{3} {(15000 - p^{3})}^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $- 1$ by $3$ .

$\frac{1}{3} {(15000 - p^{3})}^{\frac{1 - 3}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.5.2

Subtract $3$ from $1$ .

$\frac{1}{3} {(15000 - p^{3})}^{\frac{- 2}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.6

Differentiate.

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Step 4.6.1

Move the negative in front of the fraction.

$\frac{1}{3} {(15000 - p^{3})}^{- \frac{2}{3}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.6.2

Combine fractions.

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Step 4.6.2.1

Combine $\frac{1}{3}$ and ${(15000 - p^{3})}^{- \frac{2}{3}}$ .

$\frac{{(15000 - p^{3})}^{- \frac{2}{3}}}{3} \frac{d}{d x} [15000 - p^{3}]$

Step 4.6.2.2

Move ${(15000 - p^{3})}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [15000 - p^{3}]$

Step 4.6.3

By the Sum Rule, the derivative of $15000 - p^{3}$ with respect to $x$ is $\frac{d}{d x} [15000] + \frac{d}{d x} [- p^{3}]$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (\frac{d}{d x} [15000] + \frac{d}{d x} [- p^{3}])$

Step 4.6.4

Since $15000$ is constant with respect to $x$ , the derivative of $15000$ with respect to $x$ is $0$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (0 + \frac{d}{d x} [- p^{3}])$

Step 4.6.5

Add $0$ and $\frac{d}{d x} [- p^{3}]$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [- p^{3}]$

Step 4.6.6

Since $- 1$ is constant with respect to $x$ , the derivative of $- p^{3}$ with respect to $x$ is $- \frac{d}{d x} [p^{3}]$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (- \frac{d}{d x} [p^{3}])$

Step 4.7

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

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Step 4.7.1

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

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (- (\frac{d}{d u_{2}} [{u_{2}}^{3}] \frac{d}{d x} [p]))$

Step 4.7.2

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

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (- (3 {u_{2}}^{2} \frac{d}{d x} [p]))$

Step 4.7.3

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

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (- (3 p^{2} \frac{d}{d x} [p]))$

Step 4.8

Simplify terms.

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Step 4.8.1

Multiply $3$ by $- 1$ .

$\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (- 3 (p^{2} \frac{d}{d x} [p]))$

Step 4.8.2

Combine $- 3$ and $\frac{1}{3 {(15000 - p^{3})}^{\frac{2}{3}}}$ .

$\frac{- 3}{3 {(15000 - p^{3})}^{\frac{2}{3}}} (p^{2} \frac{d}{d x} [p])$

Step 4.8.3

Combine $p^{2}$ and $\frac{- 3}{3 {(15000 - p^{3})}^{\frac{2}{3}}}$ .

$\frac{p^{2} \cdot - 3}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.8.4

Move $- 3$ to the left of $p^{2}$ .

$\frac{- 3 \cdot p^{2}}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.8.5

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

$\frac{3 (- p^{2})}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.9

Cancel the common factors.

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Step 4.9.1

Factor $3$ out of $3 {(15000 - p^{3})}^{\frac{2}{3}}$ .

$\frac{3 (- p^{2})}{3 ({(15000 - p^{3})}^{\frac{2}{3}})} \frac{d}{d x} [p]$

Step 4.9.2

Cancel the common factor.

$\frac{3 (- p^{2})}{3 {(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.9.3

Rewrite the expression.

$\frac{- p^{2}}{{(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.10

Move the negative in front of the fraction.

$- \frac{p^{2}}{{(15000 - p^{3})}^{\frac{2}{3}}} \frac{d}{d x} [p]$

Step 4.11

Rewrite $\frac{d}{d x} [p]$ as $p'$ .

$- \frac{p^{2}}{{(15000 - p^{3})}^{\frac{2}{3}}} p'$

Step 4.12

Combine $p'$ and $\frac{p^{2}}{{(15000 - p^{3})}^{\frac{2}{3}}}$ .

$- \frac{p' p^{2}}{{(15000 - p^{3})}^{\frac{2}{3}}}$

Step 4.13

Reorder factors in $p' p^{2}$ .

$- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}}$

Step 5

Reform the equation by setting the left side equal to the right side.

$1 = - \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}}$

Step 6

Solve for $p'$ .

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Step 6.1

Rewrite the equation as $- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = 1$ .

$- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = 1$

Step 6.2

Divide each term in $- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = 1$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = 1$ by $- 1$ .

$\frac{- \frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}}}{- 1} = \frac{1}{- 1}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}}}{1} = \frac{1}{- 1}$

Step 6.2.2.2

Divide $\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}}$ by $1$ .

$\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = \frac{1}{- 1}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $1$ by $- 1$ .

$\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} = - 1$

Step 6.3

Multiply both sides by ${(15000 - p^{3})}^{\frac{2}{3}}$ .

$\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} {(15000 - p^{3})}^{\frac{2}{3}} = - {(15000 - p^{3})}^{\frac{2}{3}}$

Step 6.4

Simplify the left side.

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Step 6.4.1

Cancel the common factor of ${(15000 - p^{3})}^{\frac{2}{3}}$ .

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Step 6.4.1.1

Cancel the common factor.

$\frac{p^{2} p'}{{(15000 - p^{3})}^{\frac{2}{3}}} {(15000 - p^{3})}^{\frac{2}{3}} = - {(15000 - p^{3})}^{\frac{2}{3}}$

Step 6.4.1.2

Rewrite the expression.

$p^{2} p' = - {(15000 - p^{3})}^{\frac{2}{3}}$

Step 6.5

Divide each term in $p^{2} p' = - {(15000 - p^{3})}^{\frac{2}{3}}$ by $p^{2}$ and simplify.

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Step 6.5.1

Divide each term in $p^{2} p' = - {(15000 - p^{3})}^{\frac{2}{3}}$ by $p^{2}$ .

$\frac{p^{2} p'}{p^{2}} = \frac{- {(15000 - p^{3})}^{\frac{2}{3}}}{p^{2}}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $p^{2}$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{p^{2} p'}{p^{2}} = \frac{- {(15000 - p^{3})}^{\frac{2}{3}}}{p^{2}}$

Step 6.5.2.1.2

Divide $p'$ by $1$ .

$p' = \frac{- {(15000 - p^{3})}^{\frac{2}{3}}}{p^{2}}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Move the negative in front of the fraction.

$p' = - \frac{{(15000 - p^{3})}^{\frac{2}{3}}}{p^{2}}$

Step 7

Replace $p'$ with $\frac{d p}{d x}$ .

$\frac{d p}{d x} = - \frac{{(15000 - p^{3})}^{\frac{2}{3}}}{p^{2}}$