# How do you find dy/dx by implicit differentiation given e^y=x^2+y?

Dec 16, 2016

Note that since $y = y \left(x\right)$, taking the derivative of the entire function with respect to $y$ requires you to account for that fact using the chain rule.

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dy}} \frac{\mathrm{dy}}{\mathrm{dx}}$

where $\frac{\mathrm{df}}{\mathrm{dy}}$ is the derivative of, say, ${e}^{y}$ or $y$ in your given function.

Thus:

${e}^{y} \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x + 1 \frac{\mathrm{dy}}{\mathrm{dx}}$

$\implies \left({e}^{y} - 1\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x$

$\implies \textcolor{b l u e}{\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{{e}^{y} - 1}}$

If you know what $y$ is in terms of $x$, you can plug it back in to give you a function containing just $x$.