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We first consider the definition of Bessel function of order n and differentiate it w.r.t. x.
Let us now multiply both sides by x, in the above derivative of Jn. After this we add the powers of two (x/2) terms. This will help us change the power of (x/2) back to its original value.
Next, we split the summation in to two parts by opening the brackets of the term (n + 2r). By doing this, we observe that the first summation is the definition of Jn. In the second summation, we cancel out r from numerator and denominator.
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