I have shown that the Bessel function $$J_n\left(x\right)=\dfrac{x^n}{2^n}\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^kx^{2k}}{2^{2k}k!\left(n+k\right)!}$$ satisfies the property
$$\left(x^{n+1}J_{n+1}\left(x\right)\right)'=x^{n+1}J_{n}\left(x\right)$$
Hence or otherwise, show that $$J'_{n+1}\left(x\right)=J_n\left(x\right)-\dfrac{n+1}{x}J_{n+1}\left(x\right)$$
One of the basic properties of Bessel functions is $(x^{n+1}J_{n+1}(x))'=x^{n+1}J_{n}(x)$ (It should be $J_{n}(x)$ on the R.H.S of your equation). It can be easily derived by substituting $n+1$ in place of $n$ in the power series expression of $J_{n}(x)$ and differentiating w.r.t $x$. Then by expanding the L.H.S in the above equation using chain rule and rearranging, the required equation/identity can be obtained.