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EXAMPLE 9Using the Chain Rule with the Inverse Sine Function

Find y if:
(a) y=sin1(4x2) (b) y=esin1x

Solution(a) If y=sin1u and u=4x2, then dydu=11u2 and dudx=8x. By the Chain Rule, y=dydx=dydududx=(11u2)(8x)=8x116x4u=4x2

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(b) If y=eu and u=sin1x, then dydu=eu and dudx=11x2. By the Chain Rule, y=dydx=dydududx=eu11x2=u=sin1xesin1x1x2