QuestionFind the indicated partial derivative. $$f(x, y)=y \sin ^{-1}(x y) ; \quad f_{y}\left(4, \frac{1}{8}\right)$$

Studdy Solution
Simplify the expression:
$$\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$$
$$\sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$$
Substitute these values back:
$$f_y\left(4, \frac{1}{8}\right) = \frac{\pi}{6} + \frac{1}{8} \cdot \frac{4}{\frac{\sqrt{3}}{2}}$$
$$= \frac{\pi}{6} + \frac{1}{8} \cdot \frac{8}{\sqrt{3}}$$
$$= \frac{\pi}{6} + \frac{1}{\sqrt{3}}$$
The indicated partial derivative is $\boxed{\frac{\pi}{6} + \frac{1}{\sqrt{3}}}$.