Given the Gauss hypergeometric function ${}_{2}F_{1} (a,b,c,z)$, its $n$th derivative can be written as
$\frac{\mathrm{d}^{n}}{\mathrm{d} z^{n}} {}_{2}F_{1} (a,b,c,z) = \frac{(a)_{n} (b)_{n}}{(c)_{n}} {}_{2}F_{1} (a+n,b+n,c+n,z)$
However, when $c = b+1$, it appears that one can write the $n$th derivative differently. For instance, with Mathematica, I obtain
$\frac{\mathrm{d}}{\mathrm{d} z} {}_{2}F_{1} (a,b,b+1,z) = \frac{b \big( (1-z)^{-a} - {}_{2}F_{1} (a,b,b+1,z) \big)}{z}$
$\frac{\mathrm{d}^{2}}{\mathrm{d} z^{2}} {}_{2}F_{1} (a,b,b+1,z) = \frac{b \left(a (1-z)^{-a-1}-\frac{b \left((1-z)^{-a}-\, _2F_1(a,b;b+1;z)\right)}{z}\right)}{z}-\frac{b \left((1-z)^{-a}-\, _2F_1(a,b;b+1;z)\right)}{z^2}$
$\frac{\mathrm{d}^{3}}{\mathrm{d} z^{3}} {}_{2}F_{1} (a,b,b+1,z) = \ldots$
Although less compact, the interesting thing of this form is that all Gauss hypergeometric functions appearing in the derivatives are the same as the original function. What is the property behind this result? I searched all the properties of the Gauss hypergeometric function in http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/ and in the book "Table of integrals, series and products" but with no luck.
This behavior of $F(a,b;b+1;z)$ can be seen by working with the contiguous relations of the hypergeometric function. Using the notation in the linked paper we may write the derivative of the hypergeometric function as $$ \partial_zF\left({a_1,a_2\atop a_3};z\right)=\frac{a_1a_2}{a_3}\mathcal A_1\mathcal A_2\mathcal A_3F\left({a_1,a_2\atop a_3};z\right)=\frac{a_2}{z}(\mathcal A_2-\mathcal I)F\left({a_1,a_2\atop a_3};z\right), $$ which is equivalent to stating $$ \partial_zF\left({a_1,a_2\atop a_3};z\right)=\frac{a_2}{z}\left(F\left({a_1,a_2+1\atop a_3};z\right)-F\left({a_1,a_2\atop a_3};z\right)\right). $$ For your specific case, $a_2+1=a_3$ and so this relationship reduces to $$ \partial_zF\left({a_1,a_2\atop a_2+1};z\right)=\frac{a_2}{z}\left((1-z)^{-a_1}-F\left({a_1,a_2\atop a_2+1};z\right)\right). $$ Note that we can iterate this process and write higher order derivatives of $F(a,b;b+1;z)$ w.r.t. $z$ in terms of combinations of powers of $z$, $(1-z)$, and $F(a,b;b+1;z)$.