Calculus Examples

$x^{\frac{4}{3}} + y^{\frac{4}{3}} = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{\frac{4}{3}} + y^{\frac{4}{3}}) = \frac{d}{d x} (1)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x^{\frac{4}{3}} + y^{\frac{4}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [y^{\frac{4}{3}}]$ .

$\frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2

Evaluate $\frac{d}{d x} [x^{\frac{4}{3}}]$ .

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Step 2.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{4}{3} x^{\frac{4}{3} - 1} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2.2

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

$\frac{4}{3} x^{\frac{4}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2.3

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

$\frac{4}{3} x^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{4}{3} x^{\frac{4 - 1 \cdot 3}{3}} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $3$ .

$\frac{4}{3} x^{\frac{4 - 3}{3}} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.2.5.2

Subtract $3$ from $4$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{d}{d x} [y^{\frac{4}{3}}]$

Step 2.3

Evaluate $\frac{d}{d x} [y^{\frac{4}{3}}]$ .

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Step 2.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{3}}$ and $g (x) = y$ .

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Step 2.3.1.1

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

$\frac{4}{3} x^{\frac{1}{3}} + \frac{d}{d u} [u^{\frac{4}{3}}] \frac{d}{d x} [y]$

Step 2.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{3}$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} u^{\frac{4}{3} - 1} \frac{d}{d x} [y]$

Step 2.3.1.3

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

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4}{3} - 1} \frac{d}{d x} [y]$

Step 2.3.2

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

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4}{3} - 1} y'$

Step 2.3.3

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

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4}{3} - 1 \cdot \frac{3}{3}} y'$

Step 2.3.4

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

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4}{3} + \frac{- 1 \cdot 3}{3}} y'$

Step 2.3.5

Combine the numerators over the common denominator.

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4 - 1 \cdot 3}{3}} y'$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $3$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{4 - 3}{3}} y'$

Step 2.3.6.2

Subtract $3$ from $4$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4}{3} y^{\frac{1}{3}} y'$

Step 2.3.7

Combine $\frac{4}{3}$ and $y^{\frac{1}{3}}$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4 y^{\frac{1}{3}}}{3} y'$

Step 2.3.8

Combine $\frac{4 y^{\frac{1}{3}}}{3}$ and $y'$ .

$\frac{4}{3} x^{\frac{1}{3}} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 2.4

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 3

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

$0$

Step 4

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

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4 y^{\frac{1}{3}} y'}{3} = 0$

Step 5

Solve for $x^{\frac{1}{3}}$ .

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Step 5.1

Find a common factor $x^{\frac{1}{3}}$ that is present in each term.

$\frac{4 x^{\frac{1}{3}}}{3} + \frac{4 y^{\frac{1}{3}} y'}{3}$

Step 5.2

Substitute $u$ for $x^{\frac{1}{3}}$ .

$\frac{4 (u)}{3} + \frac{4 y^{\frac{1}{3}} y'}{3} = 0$

Step 5.3

Solve for $u$ .

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Step 5.3.1

Reorder factors in $\frac{4 u}{3} + \frac{4 y^{\frac{1}{3}} y'}{3}$ .

$\frac{4 u}{3} + \frac{4 y' y^{\frac{1}{3}}}{3} = 0$

Step 5.3.2

Subtract $\frac{4 y' y^{\frac{1}{3}}}{3}$ from both sides of the equation.

$\frac{4 u}{3} = - \frac{4 y' y^{\frac{1}{3}}}{3}$

Step 5.3.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$4 u = - 4 y' y^{\frac{1}{3}}$

Step 5.3.4

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

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Step 5.3.4.1

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

$\frac{4 u}{4} = \frac{- 4 y' y^{\frac{1}{3}}}{4}$

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

Cancel the common factor of $4$ .

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Step 5.3.4.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{- 4 y' y^{\frac{1}{3}}}{4}$

Step 5.3.4.2.1.2

Divide $u$ by $1$ .

$u = \frac{- 4 y' y^{\frac{1}{3}}}{4}$

Step 5.3.4.3

Simplify the right side.

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Step 5.3.4.3.1

Factor $4$ out of $- 4 y' y^{\frac{1}{3}}$ .

$u = \frac{4 (- y' y^{\frac{1}{3}})}{4}$

Step 5.3.4.3.2

Cancel the common factors.

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Step 5.3.4.3.2.1

Factor $4$ out of $4$ .

$u = \frac{4 (- y' y^{\frac{1}{3}})}{4 (1)}$

Step 5.3.4.3.2.2

Cancel the common factor.

$u = \frac{4 (- y' y^{\frac{1}{3}})}{4 \cdot 1}$

Step 5.3.4.3.2.3

Rewrite the expression.

$u = \frac{- y' y^{\frac{1}{3}}}{1}$

Step 5.3.4.3.2.4

Divide $- y' y^{\frac{1}{3}}$ by $1$ .

$u = - y' y^{\frac{1}{3}}$

Step 5.4

Substitute $y'$ for $u$ .

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

Step 6

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

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