Calculus Examples

$x^{\frac{6}{7}} + y^{\frac{8}{5}} = 9$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{\frac{6}{7}} + y^{\frac{8}{5}}) = \frac{d}{d x} (9)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $x^{\frac{6}{7}} + y^{\frac{8}{5}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{6}{7}}] + \frac{d}{d x} [y^{\frac{8}{5}}]$ .

$\frac{d}{d x} [x^{\frac{6}{7}}] + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2

Evaluate $\frac{d}{d x} [x^{\frac{6}{7}}]$ .

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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 $n = \frac{6}{7}$ .

$\frac{6}{7} x^{\frac{6}{7} - 1} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{6}{7} x^{\frac{6}{7} - 1 \cdot \frac{7}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.3

Combine $- 1$ and $\frac{7}{7}$ .

$\frac{6}{7} x^{\frac{6}{7} + \frac{- 1 \cdot 7}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.4

Combine the numerators over the common denominator.

$\frac{6}{7} x^{\frac{6 - 1 \cdot 7}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $- 1$ by $7$ .

$\frac{6}{7} x^{\frac{6 - 7}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.5.2

Subtract $7$ from $6$ .

$\frac{6}{7} x^{\frac{- 1}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.2.6

Move the negative in front of the fraction.

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{d}{d x} [y^{\frac{8}{5}}]$

Step 2.3

Evaluate $\frac{d}{d x} [y^{\frac{8}{5}}]$ .

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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 $f (x) = x^{\frac{8}{5}}$ and $g (x) = y$ .

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Step 2.3.1.1

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

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{d}{d u} [u^{\frac{8}{5}}] \frac{d}{d x} [y]$

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 $n = \frac{8}{5}$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} u^{\frac{8}{5} - 1} \frac{d}{d x} [y]$

Step 2.3.1.3

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

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8}{5} - 1} \frac{d}{d x} [y]$

Step 2.3.2

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

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8}{5} - 1} y'$

Step 2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8}{5} - 1 \cdot \frac{5}{5}} y'$

Step 2.3.4

Combine $- 1$ and $\frac{5}{5}$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8}{5} + \frac{- 1 \cdot 5}{5}} y'$

Step 2.3.5

Combine the numerators over the common denominator.

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8 - 1 \cdot 5}{5}} y'$

Step 2.3.6

Simplify the numerator.

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Step 2.3.6.1

Multiply $- 1$ by $5$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{8 - 5}{5}} y'$

Step 2.3.6.2

Subtract $5$ from $8$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8}{5} y^{\frac{3}{5}} y'$

Step 2.3.7

Combine $\frac{8}{5}$ and $y^{\frac{3}{5}}$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8 y^{\frac{3}{5}}}{5} y'$

Step 2.3.8

Combine $\frac{8 y^{\frac{3}{5}}}{5}$ and $y'$ .

$\frac{6}{7} x^{- \frac{1}{7}} + \frac{8 y^{\frac{3}{5}} y'}{5}$

Step 2.4

Simplify.

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Step 2.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6}{7} \cdot \frac{1}{x^{\frac{1}{7}}} + \frac{8 y^{\frac{3}{5}} y'}{5}$

Step 2.4.2

Multiply $\frac{6}{7}$ by $\frac{1}{x^{\frac{1}{7}}}$ .

$\frac{6}{7 x^{\frac{1}{7}}} + \frac{8 y^{\frac{3}{5}} y'}{5}$

Step 3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0$

Step 4

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

$\frac{6}{7 x^{\frac{1}{7}}} + \frac{8 y^{\frac{3}{5}} y'}{5} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Reorder factors in $\frac{6}{7 x^{\frac{1}{7}}} + \frac{8 y^{\frac{3}{5}} y'}{5}$ .

$\frac{6}{7 x^{\frac{1}{7}}} + \frac{8 y' y^{\frac{3}{5}}}{5} = 0$

Step 5.2

Subtract $\frac{6}{7 x^{\frac{1}{7}}}$ from both sides of the equation.

$\frac{8 y' y^{\frac{3}{5}}}{5} = - \frac{6}{7 x^{\frac{1}{7}}}$

Step 5.3

Multiply both sides of the equation by $\frac{5}{8}$ .

$\frac{5}{8} \cdot \frac{8 y' y^{\frac{3}{5}}}{5} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4

Simplify both sides of the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify $\frac{5}{8} \cdot \frac{8 y' y^{\frac{3}{5}}}{5}$ .

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Step 5.4.1.1.1

Combine.

$\frac{5 (8 y' y^{\frac{3}{5}})}{8 \cdot 5} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4.1.1.2

Cancel the common factor.

$\frac{5 (8 y' y^{\frac{3}{5}})}{8 \cdot 5} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4.1.1.3

Rewrite the expression.

$\frac{8 y' y^{\frac{3}{5}}}{8} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4.1.1.4

Cancel the common factor.

$\frac{8 y' y^{\frac{3}{5}}}{8} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4.1.1.5

Divide $y' y^{\frac{3}{5}}$ by $1$ .

$y' y^{\frac{3}{5}} = \frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\frac{5}{8} (- \frac{6}{7 x^{\frac{1}{7}}})$ .

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Step 5.4.2.1.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1.1

Move the leading negative in $- \frac{6}{7 x^{\frac{1}{7}}}$ into the numerator.

$y' y^{\frac{3}{5}} = \frac{5}{8} \cdot \frac{- 6}{7 x^{\frac{1}{7}}}$

Step 5.4.2.1.1.2

Factor $2$ out of $8$ .

$y' y^{\frac{3}{5}} = \frac{5}{2 (4)} \cdot \frac{- 6}{7 x^{\frac{1}{7}}}$

Step 5.4.2.1.1.3

Factor $2$ out of $- 6$ .

$y' y^{\frac{3}{5}} = \frac{5}{2 \cdot 4} \cdot \frac{2 \cdot - 3}{7 x^{\frac{1}{7}}}$

Step 5.4.2.1.1.4

Cancel the common factor.

$y' y^{\frac{3}{5}} = \frac{5}{2 \cdot 4} \cdot \frac{2 \cdot - 3}{7 x^{\frac{1}{7}}}$

Step 5.4.2.1.1.5

Rewrite the expression.

$y' y^{\frac{3}{5}} = \frac{5}{4} \cdot \frac{- 3}{7 x^{\frac{1}{7}}}$

Step 5.4.2.1.2

Multiply $\frac{5}{4}$ by $\frac{- 3}{7 x^{\frac{1}{7}}}$ .

$y' y^{\frac{3}{5}} = \frac{5 \cdot - 3}{4 (7 x^{\frac{1}{7}})}$

Step 5.4.2.1.3

Simplify the expression.

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Step 5.4.2.1.3.1

Multiply $5$ by $- 3$ .

$y' y^{\frac{3}{5}} = \frac{- 15}{4 (7 x^{\frac{1}{7}})}$

Step 5.4.2.1.3.2

Multiply $7$ by $4$ .

$y' y^{\frac{3}{5}} = \frac{- 15}{28 x^{\frac{1}{7}}}$

Step 5.4.2.1.3.3

Move the negative in front of the fraction.

$y' y^{\frac{3}{5}} = - \frac{15}{28 x^{\frac{1}{7}}}$

Step 5.5

Divide each term in $y' y^{\frac{3}{5}} = - \frac{15}{28 x^{\frac{1}{7}}}$ by $y^{\frac{3}{5}}$ and simplify.

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Step 5.5.1

Divide each term in $y' y^{\frac{3}{5}} = - \frac{15}{28 x^{\frac{1}{7}}}$ by $y^{\frac{3}{5}}$ .

$\frac{y' y^{\frac{3}{5}}}{y^{\frac{3}{5}}} = \frac{- \frac{15}{28 x^{\frac{1}{7}}}}{y^{\frac{3}{5}}}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor.

$\frac{y' y^{\frac{3}{5}}}{y^{\frac{3}{5}}} = \frac{- \frac{15}{28 x^{\frac{1}{7}}}}{y^{\frac{3}{5}}}$

Step 5.5.2.2

Divide $y'$ by $1$ .

$y' = \frac{- \frac{15}{28 x^{\frac{1}{7}}}}{y^{\frac{3}{5}}}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{15}{28 x^{\frac{1}{7}}} \cdot \frac{1}{y^{\frac{3}{5}}}$

Step 5.5.3.2

Multiply $\frac{1}{y^{\frac{3}{5}}}$ by $\frac{15}{28 x^{\frac{1}{7}}}$ .

$y' = - \frac{15}{y^{\frac{3}{5}} (28 x^{\frac{1}{7}})}$

Step 5.5.3.3

Move $28$ to the left of $y^{\frac{3}{5}}$ .

$y' = - \frac{15}{28 y^{\frac{3}{5}} x^{\frac{1}{7}}}$

Step 6

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

$\frac{d y}{d x} = - \frac{15}{28 y^{\frac{3}{5}} x^{\frac{1}{7}}}$