Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$x e^{\frac{y}{z}} (y (- z^{- 2}))$

Step 4

Simplify.

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Step 4.1

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

$x e^{\frac{y}{z}} (y (- \frac{1}{z^{2}}))$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $y$ and $\frac{1}{z^{2}}$ .

$x e^{\frac{y}{z}} (- \frac{y}{z^{2}})$

Step 4.2.2

Combine $x$ and $\frac{y}{z^{2}}$ .

$e^{\frac{y}{z}} (- \frac{x y}{z^{2}})$

Step 4.2.3

Combine $e^{\frac{y}{z}}$ and $\frac{x y}{z^{2}}$ .

$- \frac{e^{\frac{y}{z}} (x y)}{z^{2}}$

Step 4.3

Remove unnecessary parentheses.

$- \frac{e^{\frac{y}{z}} x y}{z^{2}}$

Step 4.4

Reorder factors in $- \frac{e^{\frac{y}{z}} x y}{z^{2}}$ .

$- \frac{x y e^{\frac{y}{z}}}{z^{2}}$