Calculus Examples

$x^{\frac{y}{z}}$

Step 1

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

$\frac{d}{d y} [x] (\frac{y}{z} x^{\frac{y}{z} - 1}) + \frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$

Step 2

Differentiate.

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Step 2.1

Combine $\frac{y}{z}$ and $x^{\frac{y}{z} - 1}$ .

$\frac{d}{d y} [x] \frac{y x^{\frac{y}{z} - 1}}{z} + \frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$

Step 2.2

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

$0 \frac{y x^{\frac{y}{z} - 1}}{z} + \frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $0$ by $\frac{y x^{\frac{y}{z} - 1}}{z}$ .

$0 + \frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$

Step 2.3.2

Add $0$ and $\frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$ .

$\frac{d}{d y} [\frac{y}{z}] (x^{\frac{y}{z}} \ln (x))$

Step 2.4

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

$\frac{1}{z} \frac{d}{d y} [y] (x^{\frac{y}{z}} \ln (x))$

Step 2.5

Combine fractions.

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Step 2.5.1

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

$\frac{x^{\frac{y}{z}}}{z} \frac{d}{d y} [y] \ln (x)$

Step 2.5.2

Combine $\ln (x)$ and $\frac{x^{\frac{y}{z}}}{z}$ .

$\frac{\ln (x) x^{\frac{y}{z}}}{z} \frac{d}{d y} [y]$

Step 2.6

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

$\frac{\ln (x) x^{\frac{y}{z}}}{z} \cdot 1$

Step 2.7

Simplify the expression.

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Step 2.7.1

Multiply $\frac{\ln (x) x^{\frac{y}{z}}}{z}$ by $1$ .

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

Step 2.7.2

Reorder factors in $\ln (x) x^{\frac{y}{z}}$ .

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