Calculus Examples

$\ln (\frac{x^{40} y^{30}}{z^{20}})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d z} [f (g (z))]$ is $f' (g (z)) g' (z)$ where $f (z) = \ln (z)$ and $g (z) = \frac{x^{40} y^{30}}{z^{20}}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x^{40} y^{30}}{z^{20}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d z} [\frac{x^{40} y^{30}}{z^{20}}]$

Step 1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d z} [\frac{x^{40} y^{30}}{z^{20}}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x^{40} y^{30}}{z^{20}}$ .

$\frac{1}{\frac{x^{40} y^{30}}{z^{20}}} \frac{d}{d z} [\frac{x^{40} y^{30}}{z^{20}}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x^{40} y^{30}}{z^{20}}$ .

$1 \frac{z^{20}}{x^{40} y^{30}} \frac{d}{d z} [\frac{x^{40} y^{30}}{z^{20}}]$

Step 3

Multiply $\frac{z^{20}}{x^{40} y^{30}}$ by $1$ .

$\frac{z^{20}}{x^{40} y^{30}} \frac{d}{d z} [\frac{x^{40} y^{30}}{z^{20}}]$

Step 4

Since $x^{40} y^{30}$ is constant with respect to $z$ , the derivative of $\frac{x^{40} y^{30}}{z^{20}}$ with respect to $z$ is $x^{40} y^{30} \frac{d}{d z} [\frac{1}{z^{20}}]$ .

$\frac{z^{20}}{x^{40} y^{30}} (x^{40} y^{30} \frac{d}{d z} [\frac{1}{z^{20}}])$

Step 5

Simplify terms.

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Step 5.1

Combine $x^{40}$ and $\frac{z^{20}}{x^{40} y^{30}}$ .

$\frac{x^{40} z^{20}}{x^{40} y^{30}} (y^{30} \frac{d}{d z} [\frac{1}{z^{20}}])$

Step 5.2

Combine $y^{30}$ and $\frac{x^{40} z^{20}}{x^{40} y^{30}}$ .

$\frac{y^{30} (x^{40} z^{20})}{x^{40} y^{30}} \frac{d}{d z} [\frac{1}{z^{20}}]$

Step 5.3

Cancel the common factor of $y^{30}$ .

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Step 5.3.1

Cancel the common factor.

$\frac{y^{30} x^{40} z^{20}}{x^{40} y^{30}} \frac{d}{d z} [\frac{1}{z^{20}}]$

Step 5.3.2

Rewrite the expression.

$\frac{x^{40} z^{20}}{x^{40}} \frac{d}{d z} [\frac{1}{z^{20}}]$

Step 5.4

Cancel the common factor of $x^{40}$ .

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Step 5.4.1

Cancel the common factor.

$\frac{x^{40} z^{20}}{x^{40}} \frac{d}{d z} [\frac{1}{z^{20}}]$

Step 5.4.2

Divide $z^{20}$ by $1$ .

$z^{20} \frac{d}{d z} [\frac{1}{z^{20}}]$

Step 5.5

Apply basic rules of exponents.

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Step 5.5.1

Rewrite $\frac{1}{z^{20}}$ as ${(z^{20})}^{- 1}$ .

$z^{20} \frac{d}{d z} [{(z^{20})}^{- 1}]$

Step 5.5.2

Multiply the exponents in ${(z^{20})}^{- 1}$ .

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Step 5.5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z^{20} \frac{d}{d z} [z^{20 \cdot - 1}]$

Step 5.5.2.2

Multiply $20$ by $- 1$ .

$z^{20} \frac{d}{d z} [z^{- 20}]$

Step 6

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = - 20$ .

$z^{20} (- 20 z^{- 21})$

Step 7

Multiply $z^{20}$ by $z^{- 21}$ by adding the exponents.

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Step 7.1

Move $z^{- 21}$ .

$z^{- 21} z^{20} \cdot - 20$

Step 7.2

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

$z^{- 21 + 20} \cdot - 20$

Step 7.3

Add $- 21$ and $20$ .

$z^{- 1} \cdot - 20$

Step 8

Move $- 20$ to the left of $z^{- 1}$ .

$- 20 \cdot z^{- 1}$

Step 9

Simplify.

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Step 9.1

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

$- 20 \frac{1}{z}$

Step 9.2

Combine terms.

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Step 9.2.1

Combine $- 20$ and $\frac{1}{z}$ .

$\frac{- 20}{z}$

Step 9.2.2

Move the negative in front of the fraction.

$- \frac{20}{z}$