Calculus Examples

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

Step 1

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

$x \frac{d}{d y} [e^{\frac{y}{z}}]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{y}{z}$ .

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

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

Step 2.3

Replace all occurrences of $u$ with $\frac{y}{z}$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Combine fractions.

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Step 3.2.1

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

$\frac{x}{z} e^{\frac{y}{z}} \frac{d}{d y} [y]$

Step 3.2.2

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

$\frac{x e^{\frac{y}{z}}}{z} \frac{d}{d y} [y]$

Step 3.3

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

$\frac{x e^{\frac{y}{z}}}{z} \cdot 1$

Step 3.4

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

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