# How do you differentiate cos(x/y) = x/y?

May 30, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{t} _ 1$ where ${t}_{1}$ is the unique root of $\cos \left(t\right) = t$

#### Explanation:

The equation $\cos \left(t\right) = t$ has only one solution, ${t}_{1} \approx 0.7390851332$

So our original equation amounts to $\frac{x}{y} = {t}_{1}$

Hence $y = \frac{x}{t} _ 1$

So $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{t} _ 1 \approx 1.353024$

graph{(y-cos(x))(y-x)=0 [-4.44, 5.56, -2.03, 2.97]}