Calculus Examples

$a = 5 y^{0.6} x^{- 0.3} z$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (a) = \frac{d}{d z} (5 y^{0.6} x^{- 0.3} z)$

Step 2

Since $a$ is constant with respect to $z$ , the derivative of $a$ with respect to $z$ is $0$ .

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

$\frac{d}{d z} [5 y^{0.6} \frac{1}{x^{0.3}} z]$

Step 3.2

Differentiate using the Constant Multiple Rule.

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Step 3.2.1

Combine $5$ and $\frac{1}{x^{0.3}}$ .

$\frac{d}{d z} [y^{0.6} \frac{5}{x^{0.3}} z]$

Step 3.2.2

Combine fractions.

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Step 3.2.2.1

Combine $y^{0.6}$ and $\frac{5}{x^{0.3}}$ .

$\frac{d}{d z} [\frac{y^{0.6} \cdot 5}{x^{0.3}} z]$

Step 3.2.2.2

Move $5$ to the left of $y^{0.6}$ .

$\frac{d}{d z} [\frac{5 \cdot y^{0.6}}{x^{0.3}} z]$

Step 3.2.2.3

Combine $\frac{5 y^{0.6}}{x^{0.3}}$ and $z$ .

$\frac{d}{d z} [\frac{5 y^{0.6} z}{x^{0.3}}]$

Step 3.2.3

Since $\frac{5}{x^{0.3}}$ is constant with respect to $z$ , the derivative of $\frac{5 y^{0.6} z}{x^{0.3}}$ with respect to $z$ is $\frac{5}{x^{0.3}} \frac{d}{d z} [y^{0.6} z]$ .

$\frac{5}{x^{0.3}} \frac{d}{d z} [y^{0.6} z]$

Step 3.3

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = y^{0.6}$ and $g (z) = z$ .

$\frac{5}{x^{0.3}} (y^{0.6} \frac{d}{d z} [z] + z \frac{d}{d z} [y^{0.6}])$

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

$\frac{5}{x^{0.3}} (y^{0.6} \cdot 1 + z \frac{d}{d z} [y^{0.6}])$

Step 3.4.2

Multiply $y^{0.6}$ by $1$ .

$\frac{5}{x^{0.3}} (y^{0.6} + z \frac{d}{d z} [y^{0.6}])$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = z^{0.6}$ and $g (z) = y$ .

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Step 3.5.1

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

$\frac{5}{x^{0.3}} (y^{0.6} + z (\frac{d}{d u} [u^{0.6}] \frac{d}{d z} [y]))$

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 0.6$ .

$\frac{5}{x^{0.3}} (y^{0.6} + z (0.6 u^{- 0.4} \frac{d}{d z} [y]))$

Step 3.5.3

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

$\frac{5}{x^{0.3}} (y^{0.6} + z (0.6 y^{- 0.4} \frac{d}{d z} [y]))$

Step 3.6

Move $0.6$ to the left of $z$ .

$\frac{5}{x^{0.3}} (y^{0.6} + 0.6 \cdot z y^{- 0.4} \frac{d}{d z} [y])$

Step 3.7

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

$\frac{5}{x^{0.3}} (y^{0.6} + 0.6 z y^{- 0.4} y')$

Step 3.8

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

$\frac{5}{x^{0.3}} (y^{0.6} + 0.6 z \frac{1}{y^{0.4}} y')$

Step 3.9

Simplify.

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Step 3.9.1

Apply the distributive property.

$\frac{5}{x^{0.3}} y^{0.6} + \frac{5}{x^{0.3}} (0.6 z \frac{1}{y^{0.4}} y')$

Step 3.9.2

Combine terms.

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Step 3.9.2.1

Combine $\frac{5}{x^{0.3}}$ and $y^{0.6}$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5}{x^{0.3}} (0.6 z \frac{1}{y^{0.4}} y')$

Step 3.9.2.2

Combine $0.6$ and $\frac{1}{y^{0.4}}$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5}{x^{0.3}} (z \frac{0.6}{y^{0.4}} y')$

Step 3.9.2.3

Combine $z$ and $\frac{0.6}{y^{0.4}}$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5}{x^{0.3}} (\frac{z \cdot 0.6}{y^{0.4}} y')$

Step 3.9.2.4

Move $0.6$ to the left of $z$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5}{x^{0.3}} (\frac{0.6 \cdot z}{y^{0.4}} y')$

Step 3.9.2.5

Combine $\frac{0.6 z}{y^{0.4}}$ and $y'$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5}{x^{0.3}} \cdot \frac{0.6 z y'}{y^{0.4}}$

Step 3.9.2.6

Multiply $\frac{5}{x^{0.3}}$ by $\frac{0.6 z y'}{y^{0.4}}$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{5 (0.6 z y')}{x^{0.3} y^{0.4}}$

Step 3.9.2.7

Multiply $0.6$ by $5$ .

$\frac{5 y^{0.6}}{x^{0.3}} + \frac{3 z y'}{x^{0.3} y^{0.4}}$

Step 3.9.3

Reorder terms.

$\frac{3 z y'}{x^{0.3} y^{0.4}} + \frac{5 y^{0.6}}{x^{0.3}}$

Step 4

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

$0 = \frac{3 z y'}{x^{0.3} y^{0.4}} + \frac{5 y^{0.6}}{x^{0.3}}$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $\frac{3 z y'}{x^{0.3} y^{0.4}} + \frac{5 y^{0.6}}{x^{0.3}} = 0$ .

$\frac{3 z y'}{x^{0.3} y^{0.4}} + \frac{5 y^{0.6}}{x^{0.3}} = 0$

Step 5.2

Subtract $\frac{5 y^{0.6}}{x^{0.3}}$ from both sides of the equation.

$\frac{3 z y'}{x^{0.3} y^{0.4}} = - \frac{5 y^{0.6}}{x^{0.3}}$

Step 5.3

Multiply both sides by $x^{0.3} y^{0.4}$ .

$\frac{3 z y'}{x^{0.3} y^{0.4}} (x^{0.3} y^{0.4}) = - \frac{5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Cancel the common factor of $x^{0.3} y^{0.4}$ .

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Step 5.4.1.1.1

Cancel the common factor.

$\frac{3 z y'}{x^{0.3} y^{0.4}} (x^{0.3} y^{0.4}) = - \frac{5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$

Step 5.4.1.1.2

Rewrite the expression.

$3 z y' = - \frac{5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $- \frac{5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$ .

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Step 5.4.2.1.1

Cancel the common factor of $x^{0.3}$ .

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Step 5.4.2.1.1.1

Move the leading negative in $- \frac{5 y^{0.6}}{x^{0.3}}$ into the numerator.

$3 z y' = \frac{- 5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$

Step 5.4.2.1.1.2

Factor $x^{0.3}$ out of $x^{0.3} y^{0.4}$ .

$3 z y' = \frac{- 5 y^{0.6}}{x^{0.3}} (x^{0.3} (y^{0.4}))$

Step 5.4.2.1.1.3

Cancel the common factor.

$3 z y' = \frac{- 5 y^{0.6}}{x^{0.3}} (x^{0.3} y^{0.4})$

Step 5.4.2.1.1.4

Rewrite the expression.

$3 z y' = - 5 y^{0.6} y^{0.4}$

Step 5.4.2.1.2

Multiply $y^{0.6}$ by $y^{0.4}$ by adding the exponents.

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Step 5.4.2.1.2.1

Move $y^{0.4}$ .

$3 z y' = - 5 (y^{0.4} y^{0.6})$

Step 5.4.2.1.2.2

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

$3 z y' = - 5 y^{0.4 + 0.6}$

Step 5.4.2.1.2.3

Add $0.4$ and $0.6$ .

$3 z y' = - 5 y^{1}$

Step 5.4.2.1.3

Simplify $- 5 y^{1}$ .

$3 z y' = - 5 y$

Step 5.5

Divide each term in $3 z y' = - 5 y$ by $3 z$ and simplify.

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Step 5.5.1

Divide each term in $3 z y' = - 5 y$ by $3 z$ .

$\frac{3 z y'}{3 z} = \frac{- 5 y}{3 z}$

Step 5.5.2

Simplify the left side.

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Step 5.5.2.1

Cancel the common factor of $3$ .

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Step 5.5.2.1.1

Cancel the common factor.

$\frac{3 z y'}{3 z} = \frac{- 5 y}{3 z}$

Step 5.5.2.1.2

Rewrite the expression.

$\frac{z y'}{z} = \frac{- 5 y}{3 z}$

Step 5.5.2.2

Cancel the common factor of $z$ .

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Step 5.5.2.2.1

Cancel the common factor.

$\frac{z y'}{z} = \frac{- 5 y}{3 z}$

Step 5.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 5 y}{3 z}$

Step 5.5.3

Simplify the right side.

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Step 5.5.3.1

Cancel the common factor of $- 5$ and $3$ .

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Step 5.5.3.1.1

Factor $1$ out of $- 5 y$ .

$y' = \frac{1 (- 5 y)}{3 z}$

Step 5.5.3.1.2

Cancel the common factors.

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Step 5.5.3.1.2.1

Factor $1$ out of $3 z$ .

$y' = \frac{1 (- 5 y)}{1 (3 z)}$

Step 5.5.3.1.2.2

Cancel the common factor.

$y' = \frac{1 (- 5 y)}{1 (3 z)}$

Step 5.5.3.1.2.3

Rewrite the expression.

$y' = \frac{- 5 y}{3 z}$

Step 5.5.3.2

Move the negative in front of the fraction.

$y' = - \frac{5 y}{3 z}$

Step 6

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

$\frac{d y}{d z} = - \frac{5 y}{3 z}$