Calculus Examples

$\frac{x^{4} + 6 y^{5}}{x^{5} + y^{6}}$

Step 1

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 $f (y) = x^{4} + 6 y^{5}$ and $g (y) = x^{5} + y^{6}$ .

$\frac{(x^{5} + y^{6}) \frac{d}{d y} [x^{4} + 6 y^{5}] - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{4} + 6 y^{5}$ with respect to $y$ is $\frac{d}{d y} [x^{4}] + \frac{d}{d y} [6 y^{5}]$ .

$\frac{(x^{5} + y^{6}) (\frac{d}{d y} [x^{4}] + \frac{d}{d y} [6 y^{5}]) - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.2

Since $x^{4}$ is constant with respect to $y$ , the derivative of $x^{4}$ with respect to $y$ is $0$ .

$\frac{(x^{5} + y^{6}) (0 + \frac{d}{d y} [6 y^{5}]) - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.3

Add $0$ and $\frac{d}{d y} [6 y^{5}]$ .

$\frac{(x^{5} + y^{6}) \frac{d}{d y} [6 y^{5}] - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.4

Since $6$ is constant with respect to $y$ , the derivative of $6 y^{5}$ with respect to $y$ is $6 \frac{d}{d y} [y^{5}]$ .

$\frac{(x^{5} + y^{6}) (6 \frac{d}{d y} [y^{5}]) - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.5

Move $6$ to the left of $x^{5} + y^{6}$ .

$\frac{6 \cdot (x^{5} + y^{6}) \frac{d}{d y} [y^{5}] - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 5$ .

$\frac{6 (x^{5} + y^{6}) (5 y^{4}) - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.7

Multiply $5$ by $6$ .

$\frac{30 (x^{5} + y^{6}) y^{4} - (x^{4} + 6 y^{5}) \frac{d}{d y} [x^{5} + y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.8

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

$\frac{30 (x^{5} + y^{6}) y^{4} - (x^{4} + 6 y^{5}) (\frac{d}{d y} [x^{5}] + \frac{d}{d y} [y^{6}])}{{(x^{5} + y^{6})}^{2}}$

Step 2.9

Since $x^{5}$ is constant with respect to $y$ , the derivative of $x^{5}$ with respect to $y$ is $0$ .

$\frac{30 (x^{5} + y^{6}) y^{4} - (x^{4} + 6 y^{5}) (0 + \frac{d}{d y} [y^{6}])}{{(x^{5} + y^{6})}^{2}}$

Step 2.10

Add $0$ and $\frac{d}{d y} [y^{6}]$ .

$\frac{30 (x^{5} + y^{6}) y^{4} - (x^{4} + 6 y^{5}) \frac{d}{d y} [y^{6}]}{{(x^{5} + y^{6})}^{2}}$

Step 2.11

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 6$ .

$\frac{30 (x^{5} + y^{6}) y^{4} - (x^{4} + 6 y^{5}) (6 y^{5})}{{(x^{5} + y^{6})}^{2}}$

Step 2.12

Multiply $6$ by $- 1$ .

$\frac{30 (x^{5} + y^{6}) y^{4} - 6 (x^{4} + 6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(30 x^{5} + 30 y^{6}) y^{4} - 6 (x^{4} + 6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{30 x^{5} y^{4} + 30 y^{6} y^{4} - 6 (x^{4} + 6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{30 x^{5} y^{4} + 30 y^{6} y^{4} + (- 6 x^{4} - 6 (6 y^{5})) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.4

Apply the distributive property.

$\frac{30 x^{5} y^{4} + 30 y^{6} y^{4} - 6 x^{4} y^{5} - 6 (6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Multiply $y^{6}$ by $y^{4}$ by adding the exponents.

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Step 3.5.1.1.1

Move $y^{4}$ .

$\frac{30 x^{5} y^{4} + 30 (y^{4} y^{6}) - 6 x^{4} y^{5} - 6 (6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.1.2

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

$\frac{30 x^{5} y^{4} + 30 y^{4 + 6} - 6 x^{4} y^{5} - 6 (6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.1.3

Add $4$ and $6$ .

$\frac{30 x^{5} y^{4} + 30 y^{10} - 6 x^{4} y^{5} - 6 (6 y^{5}) y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.2

Multiply $y^{5}$ by $y^{5}$ by adding the exponents.

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Step 3.5.1.2.1

Move $y^{5}$ .

$\frac{30 x^{5} y^{4} + 30 y^{10} - 6 x^{4} y^{5} - 6 (6 (y^{5} y^{5}))}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.2.2

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

$\frac{30 x^{5} y^{4} + 30 y^{10} - 6 x^{4} y^{5} - 6 (6 y^{5 + 5})}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.2.3

Add $5$ and $5$ .

$\frac{30 x^{5} y^{4} + 30 y^{10} - 6 x^{4} y^{5} - 6 (6 y^{10})}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.1.3

Multiply $6$ by $- 6$ .

$\frac{30 x^{5} y^{4} + 30 y^{10} - 6 x^{4} y^{5} - 36 y^{10}}{{(x^{5} + y^{6})}^{2}}$

Step 3.5.2

Subtract $36 y^{10}$ from $30 y^{10}$ .

$\frac{30 x^{5} y^{4} - 6 y^{10} - 6 x^{4} y^{5}}{{(x^{5} + y^{6})}^{2}}$

Step 3.6

Reorder terms.

$\frac{- 6 y^{10} + 30 x^{5} y^{4} - 6 y^{5} x^{4}}{{(y^{6} + x^{5})}^{2}}$

Step 3.7

Factor $6 y^{4}$ out of $- 6 y^{10} + 30 x^{5} y^{4} - 6 y^{5} x^{4}$ .

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Step 3.7.1

Factor $6 y^{4}$ out of $- 6 y^{10}$ .

$\frac{6 y^{4} (- y^{6}) + 30 x^{5} y^{4} - 6 y^{5} x^{4}}{{(y^{6} + x^{5})}^{2}}$

Step 3.7.2

Factor $6 y^{4}$ out of $30 x^{5} y^{4}$ .

$\frac{6 y^{4} (- y^{6}) + 6 y^{4} (5 x^{5}) - 6 y^{5} x^{4}}{{(y^{6} + x^{5})}^{2}}$

Step 3.7.3

Factor $6 y^{4}$ out of $- 6 y^{5} x^{4}$ .

$\frac{6 y^{4} (- y^{6}) + 6 y^{4} (5 x^{5}) + 6 y^{4} (- y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.7.4

Factor $6 y^{4}$ out of $6 y^{4} (- y^{6}) + 6 y^{4} (5 x^{5})$ .

$\frac{6 y^{4} (- y^{6} + 5 x^{5}) + 6 y^{4} (- y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.7.5

Factor $6 y^{4}$ out of $6 y^{4} (- y^{6} + 5 x^{5}) + 6 y^{4} (- y x^{4})$ .

$\frac{6 y^{4} (- y^{6} + 5 x^{5} - y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.8

Factor $- 1$ out of $- y^{6}$ .

$\frac{6 y^{4} (- (y^{6}) + 5 x^{5} - y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.9

Factor $- 1$ out of $5 x^{5}$ .

$\frac{6 y^{4} (- (y^{6}) - (- 5 x^{5}) - y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.10

Factor $- 1$ out of $- (y^{6}) - (- 5 x^{5})$ .

$\frac{6 y^{4} (- (y^{6} - 5 x^{5}) - y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.11

Factor $- 1$ out of $- y x^{4}$ .

$\frac{6 y^{4} (- (y^{6} - 5 x^{5}) - (y x^{4}))}{{(y^{6} + x^{5})}^{2}}$

Step 3.12

Factor $- 1$ out of $- (y^{6} - 5 x^{5}) - (y x^{4})$ .

$\frac{6 y^{4} (- (y^{6} - 5 x^{5} + y x^{4}))}{{(y^{6} + x^{5})}^{2}}$

Step 3.13

Rewrite $- (y^{6} - 5 x^{5} + y x^{4})$ as $- 1 (y^{6} - 5 x^{5} + y x^{4})$ .

$\frac{6 y^{4} (- 1 (y^{6} - 5 x^{5} + y x^{4}))}{{(y^{6} + x^{5})}^{2}}$

Step 3.14

Move the negative in front of the fraction.

$- \frac{(6 y^{4}) (y^{6} - 5 x^{5} + y x^{4})}{{(y^{6} + x^{5})}^{2}}$

Step 3.15

Reorder factors in $- \frac{(6 y^{4}) (y^{6} - 5 x^{5} + y x^{4})}{{(y^{6} + x^{5})}^{2}}$ .

$- \frac{6 y^{4} (y^{6} - 5 x^{5} + y x^{4})}{{(y^{6} + x^{5})}^{2}}$