# Question #11a51

Mar 8, 2017

$y = \frac{\pi}{4} - \frac{x}{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{2}$

#### Explanation:

We could determine $\frac{\mathrm{dy}}{\mathrm{dx}}$ using implicit differentiation, but we can also note that:

$\cos \left(x + 2 y\right) = 0 \implies \left(x + 2 y\right) = \frac{\pi}{2} + k \pi$ with $k \in \mathbb{Z}$

and then:

$y = \frac{1}{2} \left(\frac{\pi}{2} + k \pi - x\right)$

So whatever is the value of $k$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{2}$

We can then also determine $k$ from:

$y \left(\frac{\pi}{6}\right) = \frac{\pi}{6}$

$\frac{1}{2} \left(\frac{\pi}{2} + k \pi - \frac{\pi}{6}\right) = \frac{\pi}{6}$

$\frac{\pi}{2} + k \pi - \frac{\pi}{6} = \frac{\pi}{3} \implies k = 0$

So in conclusion:

$y = \frac{\pi}{4} - \frac{x}{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{1}{2}$