Calculus Examples

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

Step 1

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 $g (y) = \frac{x}{y}$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Combine terms.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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