Calculus Examples

$p (x) = 2300 x^{\frac{1}{5}} y^{\frac{4}{5}}$

Step 1

Multiply $p$ by $x$ .

$p x = 2300 x^{\frac{1}{5}} y^{\frac{4}{5}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (p x) = \frac{d}{d x} (2300 x^{\frac{1}{5}} y^{\frac{4}{5}})$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

$p \frac{d}{d x} [x]$

Step 3.2

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

$p \cdot 1$

Step 3.3

Multiply $p$ by $1$ .

$p$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Since $2300$ is constant with respect to $x$ , the derivative of $2300 x^{\frac{1}{5}} y^{\frac{4}{5}}$ with respect to $x$ is $2300 \frac{d}{d x} [x^{\frac{1}{5}} y^{\frac{4}{5}}]$ .

$2300 \frac{d}{d x} [x^{\frac{1}{5}} y^{\frac{4}{5}}]$

Step 4.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{5}}$ and $g (x) = y^{\frac{4}{5}}$ .

$2300 (x^{\frac{1}{5}} \frac{d}{d x} [y^{\frac{4}{5}}] + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{5}}$ and $g (x) = y$ .

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Step 4.3.1

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

$2300 (x^{\frac{1}{5}} (\frac{d}{d u} [u^{\frac{4}{5}}] \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{5}$ .

$2300 (x^{\frac{1}{5}} (\frac{4}{5} u^{\frac{4}{5} - 1} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.3.3

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

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{4}{5} - 1} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.4

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

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{4}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.5

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

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.6

Combine the numerators over the common denominator.

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{4 - 1 \cdot 5}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $5$ .

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{4 - 5}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.7.2

Subtract $5$ from $4$ .

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{\frac{- 1}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.8

Move the negative in front of the fraction.

$2300 (x^{\frac{1}{5}} (\frac{4}{5} y^{- \frac{1}{5}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.9

Combine $\frac{4}{5}$ and $y^{- \frac{1}{5}}$ .

$2300 (x^{\frac{1}{5}} (\frac{4 y^{- \frac{1}{5}}}{5} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.10

Move $y^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2300 (x^{\frac{1}{5}} (\frac{4}{5 y^{\frac{1}{5}}} \frac{d}{d x} [y]) + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.11

Combine $\frac{4}{5 y^{\frac{1}{5}}}$ and $x^{\frac{1}{5}}$ .

$2300 (\frac{4 x^{\frac{1}{5}}}{5 y^{\frac{1}{5}}} \frac{d}{d x} [y] + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.12

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

$2300 (\frac{4 x^{\frac{1}{5}}}{5 y^{\frac{1}{5}}} y' + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.13

Combine $\frac{4 x^{\frac{1}{5}}}{5 y^{\frac{1}{5}}}$ and $y'$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} \frac{d}{d x} [x^{\frac{1}{5}}])$

Step 4.14

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

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1}))$

Step 4.15

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

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}}))$

Step 4.16

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

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}}))$

Step 4.17

Combine the numerators over the common denominator.

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}}))$

Step 4.18

Simplify the numerator.

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Step 4.18.1

Multiply $- 1$ by $5$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{1 - 5}{5}}))$

Step 4.18.2

Subtract $5$ from $1$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{\frac{- 4}{5}}))$

Step 4.19

Move the negative in front of the fraction.

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} (\frac{1}{5} x^{- \frac{4}{5}}))$

Step 4.20

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + y^{\frac{4}{5}} \frac{x^{- \frac{4}{5}}}{5})$

Step 4.21

Combine $y^{\frac{4}{5}}$ and $\frac{x^{- \frac{4}{5}}}{5}$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + \frac{y^{\frac{4}{5}} x^{- \frac{4}{5}}}{5})$

Step 4.22

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2300 (\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}})$

Step 4.23

Simplify.

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Step 4.23.1

Apply the distributive property.

$2300 \frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2

Combine terms.

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Step 4.23.2.1

Combine $2300$ and $\frac{4 x^{\frac{1}{5}} y'}{5 y^{\frac{1}{5}}}$ .

$\frac{2300 (4 x^{\frac{1}{5}} y')}{5 y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.2

Multiply $4$ by $2300$ .

$\frac{9200 (x^{\frac{1}{5}} y')}{5 y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.3

Factor $5$ out of $9200 x^{\frac{1}{5}} y'$ .

$\frac{5 (1840 x^{\frac{1}{5}} y')}{5 y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.4

Cancel the common factors.

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Step 4.23.2.4.1

Factor $5$ out of $5 y^{\frac{1}{5}}$ .

$\frac{5 (1840 x^{\frac{1}{5}} y')}{5 (y^{\frac{1}{5}})} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.4.2

Cancel the common factor.

$\frac{5 (1840 x^{\frac{1}{5}} y')}{5 y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.4.3

Rewrite the expression.

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + 2300 \frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.5

Combine $2300$ and $\frac{y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$ .

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{2300 y^{\frac{4}{5}}}{5 x^{\frac{4}{5}}}$

Step 4.23.2.6

Factor $5$ out of $2300 y^{\frac{4}{5}}$ .

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{5 (460 y^{\frac{4}{5}})}{5 x^{\frac{4}{5}}}$

Step 4.23.2.7

Cancel the common factors.

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Step 4.23.2.7.1

Factor $5$ out of $5 x^{\frac{4}{5}}$ .

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{5 (460 y^{\frac{4}{5}})}{5 (x^{\frac{4}{5}})}$

Step 4.23.2.7.2

Cancel the common factor.

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{5 (460 y^{\frac{4}{5}})}{5 x^{\frac{4}{5}}}$

Step 4.23.2.7.3

Rewrite the expression.

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}}$

Step 5

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

$p = \frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}}$

Step 6

Solve for $y'$ .

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Step 6.1

Rewrite the equation as $\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} = p$ .

$\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} = p$

Step 6.2

Reorder factors in $\frac{1840 x^{\frac{1}{5}} y'}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} - p$ .

$\frac{1840 y' x^{\frac{1}{5}}}{y^{\frac{1}{5}}} + \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} - p = 0$

Step 6.3

Move all terms not containing $y'$ to the right side of the equation.

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Step 6.3.1

Subtract $\frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}}$ from both sides of the equation.

$\frac{1840 y' x^{\frac{1}{5}}}{y^{\frac{1}{5}}} - p = - \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}}$

Step 6.3.2

Add $p$ to both sides of the equation.

$\frac{1840 y' x^{\frac{1}{5}}}{y^{\frac{1}{5}}} = - \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} + p$

Step 6.4

Multiply both sides by $y^{\frac{1}{5}}$ .

$\frac{1840 y' x^{\frac{1}{5}}}{y^{\frac{1}{5}}} y^{\frac{1}{5}} = (- \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} + p) y^{\frac{1}{5}}$

Step 6.5

Simplify.

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Step 6.5.1

Simplify the left side.

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Step 6.5.1.1

Cancel the common factor of $y^{\frac{1}{5}}$ .

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Step 6.5.1.1.1

Cancel the common factor.

$\frac{1840 y' x^{\frac{1}{5}}}{y^{\frac{1}{5}}} y^{\frac{1}{5}} = (- \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} + p) y^{\frac{1}{5}}$

Step 6.5.1.1.2

Rewrite the expression.

$1840 y' x^{\frac{1}{5}} = (- \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} + p) y^{\frac{1}{5}}$

Step 6.5.2

Simplify the right side.

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Step 6.5.2.1

Simplify $(- \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} + p) y^{\frac{1}{5}}$ .

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Step 6.5.2.1.1

Apply the distributive property.

$1840 y' x^{\frac{1}{5}} = - \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} y^{\frac{1}{5}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2

Multiply $- \frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}} y^{\frac{1}{5}}$ .

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Step 6.5.2.1.2.1

Combine $y^{\frac{1}{5}}$ and $\frac{460 y^{\frac{4}{5}}}{x^{\frac{4}{5}}}$ .

$1840 y' x^{\frac{1}{5}} = - \frac{y^{\frac{1}{5}} (460 y^{\frac{4}{5}})}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.2

Multiply $y^{\frac{1}{5}}$ by $y^{\frac{4}{5}}$ by adding the exponents.

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Step 6.5.2.1.2.2.1

Move $y^{\frac{4}{5}}$ .

$1840 y' x^{\frac{1}{5}} = - \frac{y^{\frac{4}{5}} y^{\frac{1}{5}} \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.2.2

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

$1840 y' x^{\frac{1}{5}} = - \frac{y^{\frac{4}{5} + \frac{1}{5}} \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.2.3

Combine the numerators over the common denominator.

$1840 y' x^{\frac{1}{5}} = - \frac{y^{\frac{4 + 1}{5}} \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.2.4

Add $4$ and $1$ .

$1840 y' x^{\frac{1}{5}} = - \frac{y^{\frac{5}{5}} \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.2.5

Divide $5$ by $5$ .

$1840 y' x^{\frac{1}{5}} = - \frac{y^{1} \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.2.3

Simplify $y^{1} \cdot 460$ .

$1840 y' x^{\frac{1}{5}} = - \frac{y \cdot 460}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.5.2.1.3

Move $460$ to the left of $y$ .

$1840 y' x^{\frac{1}{5}} = - \frac{460 y}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$

Step 6.6

Divide each term in $1840 y' x^{\frac{1}{5}} = - \frac{460 y}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$ by $1840 x^{\frac{1}{5}}$ and simplify.

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Step 6.6.1

Divide each term in $1840 y' x^{\frac{1}{5}} = - \frac{460 y}{x^{\frac{4}{5}}} + p y^{\frac{1}{5}}$ by $1840 x^{\frac{1}{5}}$ .

$\frac{1840 y' x^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}} = \frac{- \frac{460 y}{x^{\frac{4}{5}}}}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.2

Simplify the left side.

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Step 6.6.2.1

Cancel the common factor.

$\frac{1840 y' x^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}} = \frac{- \frac{460 y}{x^{\frac{4}{5}}}}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.2.2

Rewrite the expression.

$\frac{y' x^{\frac{1}{5}}}{x^{\frac{1}{5}}} = \frac{- \frac{460 y}{x^{\frac{4}{5}}}}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.2.3

Cancel the common factor.

$\frac{y' x^{\frac{1}{5}}}{x^{\frac{1}{5}}} = \frac{- \frac{460 y}{x^{\frac{4}{5}}}}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.2.4

Divide $y'$ by $1$ .

$y' = \frac{- \frac{460 y}{x^{\frac{4}{5}}}}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3

Simplify the right side.

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Step 6.6.3.1

Simplify each term.

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Step 6.6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y' = - \frac{460 y}{x^{\frac{4}{5}}} \cdot \frac{1}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.2

Cancel the common factor of $460$ .

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Step 6.6.3.1.2.1

Move the leading negative in $- \frac{460 y}{x^{\frac{4}{5}}}$ into the numerator.

$y' = \frac{- 460 y}{x^{\frac{4}{5}}} \cdot \frac{1}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.2.2

Factor $460$ out of $- 460 y$ .

$y' = \frac{460 (- y)}{x^{\frac{4}{5}}} \cdot \frac{1}{1840 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.2.3

Factor $460$ out of $1840 x^{\frac{1}{5}}$ .

$y' = \frac{460 (- y)}{x^{\frac{4}{5}}} \cdot \frac{1}{460 (4 x^{\frac{1}{5}})} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.2.4

Cancel the common factor.

$y' = \frac{460 (- y)}{x^{\frac{4}{5}}} \cdot \frac{1}{460 (4 x^{\frac{1}{5}})} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.2.5

Rewrite the expression.

$y' = \frac{- y}{x^{\frac{4}{5}}} \cdot \frac{1}{4 x^{\frac{1}{5}}} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.3

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

$y' = \frac{- y}{x^{\frac{4}{5}} (4 x^{\frac{1}{5}})} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.4

Multiply $x^{\frac{4}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 6.6.3.1.4.1

Move $x^{\frac{1}{5}}$ .

$y' = \frac{- y}{x^{\frac{1}{5}} x^{\frac{4}{5}} \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.4.2

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

$y' = \frac{- y}{x^{\frac{1}{5} + \frac{4}{5}} \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.4.3

Combine the numerators over the common denominator.

$y' = \frac{- y}{x^{\frac{1 + 4}{5}} \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.4.4

Add $1$ and $4$ .

$y' = \frac{- y}{x^{\frac{5}{5}} \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.4.5

Divide $5$ by $5$ .

$y' = \frac{- y}{x^{1} \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.5

Simplify $x^{1} \cdot 4$ .

$y' = \frac{- y}{x \cdot 4} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.6

Move $4$ to the left of $x$ .

$y' = \frac{- y}{4 \cdot x} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 6.6.3.1.7

Move the negative in front of the fraction.

$y' = - \frac{y}{4 x} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$

Step 7

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

$\frac{d y}{d x} = - \frac{y}{4 x} + \frac{p y^{\frac{1}{5}}}{1840 x^{\frac{1}{5}}}$