Find y′ if:
(a) y=sin−1(4x2) (b) y=esin−1x
Solution (a) If y=sin−1u and u=4x2, then dydu=1√1−u2 and dudx=8x. By the Chain Rule, y′=dydx=dydu⋅dudx=(1√1−u2)(8x)=8x√1−16x4u=4x2
217
(b) If y=eu and u=sin−1x, then dydu=eu and dudx=1√1−x2. By the Chain Rule, y′=dydx=dydu⋅dudx=eu⋅1√1−x2=↑u=sin−1xesin−1x√1−x2