Calculus Examples

$y = \frac{765}{\sqrt[3]{x}}$

Step 1

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

$y = \frac{765}{x^{\frac{1}{3}}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{765}{x^{\frac{1}{3}}})$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $765$ is constant with respect to $y$ , the derivative of $\frac{765}{x^{\frac{1}{3}}}$ with respect to $y$ is $765 \frac{d}{d y} [\frac{1}{x^{\frac{1}{3}}}]$ .

$765 \frac{d}{d y} [\frac{1}{x^{\frac{1}{3}}}]$

Step 4.2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 1$ and $g (y) = x^{\frac{1}{3}}$ .

$765 \frac{x^{\frac{1}{3}} \frac{d}{d y} [1] - 1 \cdot 1 \frac{d}{d y} [x^{\frac{1}{3}}]}{{(x^{\frac{1}{3}})}^{2}}$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Multiply $- 1$ by $1$ .

$765 \frac{x^{\frac{1}{3}} \frac{d}{d y} [1] - \frac{d}{d y} [x^{\frac{1}{3}}]}{{(x^{\frac{1}{3}})}^{2}}$

Step 4.3.2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 4.3.2.1

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

$765 \frac{x^{\frac{1}{3}} \frac{d}{d y} [1] - \frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{1}{3} \cdot 2}}$

Step 4.3.2.2

Combine $\frac{1}{3}$ and $2$ .

$765 \frac{x^{\frac{1}{3}} \frac{d}{d y} [1] - \frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}}$

Step 4.3.3

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$765 \frac{x^{\frac{1}{3}} \cdot 0 - \frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}}$

Step 4.3.4

Simplify the expression.

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Step 4.3.4.1

Multiply $x^{\frac{1}{3}}$ by $0$ .

$765 \frac{0 - \frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}}$

Step 4.3.4.2

Subtract $\frac{d}{d y} [x^{\frac{1}{3}}]$ from $0$ .

$765 \frac{- \frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}}$

Step 4.3.4.3

Move the negative in front of the fraction.

$765 (- \frac{\frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}})$

Step 4.3.4.4

Multiply $- 1$ by $765$ .

$- 765 \frac{\frac{d}{d y} [x^{\frac{1}{3}}]}{x^{\frac{2}{3}}}$

Step 4.4

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

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Step 4.4.1

To apply the Chain Rule, set $u$ as $x$ .

$- 765 \frac{\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.4.2

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

$- 765 \frac{\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.4.3

Replace all occurrences of $u$ with $x$ .

$- 765 \frac{\frac{1}{3} x^{\frac{1}{3} - 1} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.5

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

$- 765 \frac{\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.6

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

$- 765 \frac{\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.7

Combine the numerators over the common denominator.

$- 765 \frac{\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.8

Simplify the numerator.

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Step 4.8.1

Multiply $- 1$ by $3$ .

$- 765 \frac{\frac{1}{3} x^{\frac{1 - 3}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.8.2

Subtract $3$ from $1$ .

$- 765 \frac{\frac{1}{3} x^{\frac{- 2}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.9

Move the negative in front of the fraction.

$- 765 \frac{\frac{1}{3} x^{- \frac{2}{3}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.10

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

$- 765 \frac{\frac{x^{- \frac{2}{3}}}{3} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.11

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

$- 765 \frac{\frac{1}{3 x^{\frac{2}{3}}} \frac{d}{d y} [x]}{x^{\frac{2}{3}}}$

Step 4.12

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

$- 765 \frac{\frac{1}{3 x^{\frac{2}{3}}} x'}{x^{\frac{2}{3}}}$

Step 4.13

Combine $\frac{1}{3 x^{\frac{2}{3}}}$ and $x'$ .

$- 765 \frac{\frac{x'}{3 x^{\frac{2}{3}}}}{x^{\frac{2}{3}}}$

Step 4.14

Rewrite $\frac{\frac{x'}{3 x^{\frac{2}{3}}}}{x^{\frac{2}{3}}}$ as a product.

$- 765 (\frac{x'}{3 x^{\frac{2}{3}}} \cdot \frac{1}{x^{\frac{2}{3}}})$

Step 4.15

Multiply $\frac{x'}{3 x^{\frac{2}{3}}}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

$- 765 \frac{x'}{3 x^{\frac{2}{3}} x^{\frac{2}{3}}}$

Step 4.16

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

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Step 4.16.1

Move $x^{\frac{2}{3}}$ .

$- 765 \frac{x'}{3 (x^{\frac{2}{3}} x^{\frac{2}{3}})}$

Step 4.16.2

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

$- 765 \frac{x'}{3 x^{\frac{2}{3} + \frac{2}{3}}}$

Step 4.16.3

Combine the numerators over the common denominator.

$- 765 \frac{x'}{3 x^{\frac{2 + 2}{3}}}$

Step 4.16.4

Add $2$ and $2$ .

$- 765 \frac{x'}{3 x^{\frac{4}{3}}}$

Step 4.17

Combine $- 765$ and $\frac{x'}{3 x^{\frac{4}{3}}}$ .

$\frac{- 765 x'}{3 x^{\frac{4}{3}}}$

Step 4.18

Factor $3$ out of $- 765 x'$ .

$\frac{3 (- 255 x')}{3 x^{\frac{4}{3}}}$

Step 4.19

Cancel the common factors.

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Step 4.19.1

Factor $3$ out of $3 x^{\frac{4}{3}}$ .

$\frac{3 (- 255 x')}{3 (x^{\frac{4}{3}})}$

Step 4.19.2

Cancel the common factor.

$\frac{3 (- 255 x')}{3 x^{\frac{4}{3}}}$

Step 4.19.3

Rewrite the expression.

$\frac{- 255 x'}{x^{\frac{4}{3}}}$

Step 4.20

Move the negative in front of the fraction.

$- \frac{255 x'}{x^{\frac{4}{3}}}$

Step 5

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

$1 = - \frac{255 x'}{x^{\frac{4}{3}}}$

Step 6

Solve for $x'$ .

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Step 6.1

Rewrite the equation as $- \frac{255 x'}{x^{\frac{4}{3}}} = 1$ .

$- \frac{255 x'}{x^{\frac{4}{3}}} = 1$

Step 6.2

Divide each term in $- \frac{255 x'}{x^{\frac{4}{3}}} = 1$ by $- 1$ and simplify.

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Step 6.2.1

Divide each term in $- \frac{255 x'}{x^{\frac{4}{3}}} = 1$ by $- 1$ .

$\frac{- \frac{255 x'}{x^{\frac{4}{3}}}}{- 1} = \frac{1}{- 1}$

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.

$\frac{\frac{255 x'}{x^{\frac{4}{3}}}}{1} = \frac{1}{- 1}$

Step 6.2.2.2

Divide $\frac{255 x'}{x^{\frac{4}{3}}}$ by $1$ .

$\frac{255 x'}{x^{\frac{4}{3}}} = \frac{1}{- 1}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Divide $1$ by $- 1$ .

$\frac{255 x'}{x^{\frac{4}{3}}} = - 1$

Step 6.3

Multiply both sides by $x^{\frac{4}{3}}$ .

$\frac{255 x'}{x^{\frac{4}{3}}} x^{\frac{4}{3}} = - x^{\frac{4}{3}}$

Step 6.4

Simplify the left side.

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Step 6.4.1

Cancel the common factor of $x^{\frac{4}{3}}$ .

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Step 6.4.1.1

Cancel the common factor.

$\frac{255 x'}{x^{\frac{4}{3}}} x^{\frac{4}{3}} = - x^{\frac{4}{3}}$

Step 6.4.1.2

Rewrite the expression.

$255 x' = - x^{\frac{4}{3}}$

Step 6.5

Divide each term in $255 x' = - x^{\frac{4}{3}}$ by $255$ and simplify.

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Step 6.5.1

Divide each term in $255 x' = - x^{\frac{4}{3}}$ by $255$ .

$\frac{255 x'}{255} = \frac{- x^{\frac{4}{3}}}{255}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $255$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{255 x'}{255} = \frac{- x^{\frac{4}{3}}}{255}$

Step 6.5.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- x^{\frac{4}{3}}}{255}$

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.

$x' = - \frac{x^{\frac{4}{3}}}{255}$

Step 7

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

$\frac{d x}{d y} = - \frac{x^{\frac{4}{3}}}{255}$