# How do you find (dy)/(dx) given e^y=e^(3x)+5?

Nov 2, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {e}^{3 x}}{{e}^{3 x} + 5}$

#### Explanation:

To differentiate implicitly we use an implicit application of the chain rule. We can't differentiate a function of $y$ wrt $x$ but we can differentiate it wrt $y$.

So, ${e}^{y} = {e}^{3 x} + 5$

Differentiate wrt $x$:
So, $\frac{d}{\mathrm{dx}} {e}^{y} = 3 {e}^{3 x}$

And, with practice you should be able to skip this line:
$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} \frac{d}{\mathrm{dy}} {e}^{y} = 3 {e}^{3 x}$ (by the chain rule)

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} {e}^{y} = 3 {e}^{3 x}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {e}^{3 x}}{e} ^ y$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {e}^{3 x}}{{e}^{3 x} + 5}$