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The Bessel function of order 0, $ y = J (x), $ satisfies the differential equation $ xy" + y' + xy = 0 $ for all values of $ x $ and its value at 0 is $ J(0) = 1. $

(a) Find $ J'(0). $

(b) Use implicit differentiation to find $ J'(0). $

(a) $$

J^{\prime}(0)=0

$$

(b) $$

J^{\prime \prime}(0)=-\frac{1}{2}

$$

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