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I have the function of alpha + beta log(gamma((alpha+beta))), I want to calculate the derivative this function with respect to alpha. I need it as a derivative of inverted beta-liouville function with respect to each of its parameters.enter image description here

user13612169
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    Can you work with a numerical approximation or do you need the actual analytical solution? If the numerical approximation is OK, doing a (cubic) spline interpolation for example can be much simpler to derive. – PC1 Jun 14 '22 at 15:39
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    Welcome to MSE! Please learn to use MathJax so you can write formulas: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – Levon Minasian Jun 14 '22 at 15:50
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    The derivative of digamma is the trigamma function. Use the sum rule and chain rule to complete the solution. – whuber Jun 14 '22 at 15:59
  • You can use JAX to compute the derivatives: https://jax.readthedocs.io/en/latest/_autosummary/jax.scipy.special.digamma.html – Hyperplane Jun 14 '22 at 16:59
  • @PC1 I actually need its general formula not as a values, cause I want to calculate the maximum likelihood with respect to each parameter – user13612169 Jun 14 '22 at 18:42
  • In your equation, is gamma just some third variable (after alpha and beta)? The other comments seem to think it is Euler's $\Gamma$-function. – GEdgar Jun 14 '22 at 18:44
  • @whuber can you please declare it a bit, in fact the function is: log(gamma(alhpha+beta)), I know the derivative of log(gamma(x)) is digamma, so now the answer is just digamma(alpha+beta)? – user13612169 Jun 14 '22 at 18:44
  • You have neglected to differentiate the additive $\alpha$ term in your (new, edited) expression. – whuber Jun 14 '22 at 19:02
  • @GEdgar gamma is not a variable as you mentioned its a Γ(x) function. log(Γ(alpha+beta)) – user13612169 Jun 14 '22 at 22:57

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