# How do you differentiate e^y=e^(5x)-y^2?

Jul 29, 2016

I found: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5 {e}^{5 x}}{{e}^{y} + 2 y}$

#### Explanation:

Here you have an Implicit Differentiation where $y$ must be considered as a function of $x$ and differentiated as well including the term $\frac{\mathrm{dy}}{\mathrm{dx}}$.
We get:
$1 \cdot {e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{5 x} - 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$
${e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} + 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{5 x}$
collect: $\frac{\mathrm{dy}}{\mathrm{dx}}$
$\left[{e}^{y} + 2 y\right] \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = 5 {e}^{5 x}$
and finally:
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{5 {e}^{5 x}}{{e}^{y} + 2 y}$