# How do you differentiate e^(x/y)?

Sep 16, 2015

$\frac{d}{\mathrm{dx}} \left({e}^{\frac{x}{y}}\right) = = {3}^{\frac{x}{y}} \frac{y - x \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 2$

#### Explanation:

We use the derivative of the exponential, the chain rule and the quotient rule.

Assuming that we are to differentiate with respect to $x$, we get

$\frac{d}{\mathrm{dx}} \left({e}^{\frac{x}{y}}\right) = {3}^{\frac{x}{y}} \frac{d}{\mathrm{dx}} \left(\frac{x}{y}\right)$

$= {3}^{\frac{x}{y}} \frac{y - x \frac{\mathrm{dy}}{\mathrm{dx}}}{y} ^ 2$

We may prefer a different form, but that is all we can do without more information about how $x$ and $y$ are related.

If differentiating with respect to $t$ the only real difference is that $\frac{d}{\mathrm{dt}} \left(x\right)$ may not be $1$, so we write:

$\frac{d}{\mathrm{dt}} \left({e}^{\frac{x}{y}}\right) = {3}^{\frac{x}{y}} \frac{d}{\mathrm{dt}} \left(\frac{x}{y}\right) = {3}^{\frac{x}{y}} \frac{y \frac{\mathrm{dx}}{\mathrm{dt}} - x \frac{\mathrm{dy}}{\mathrm{dt}}}{y} ^ 2$