<s>[INST] derivative with quotient rule and summations I want to differentiate this with respect to $\eta$:
$$C(\eta) = \frac{1+\sum\limits_{k=1}^{3} B_k\left(\frac{\eta}{\eta_c}\right)^k}{3\left(1-\frac{\eta}{\eta_c}\right)}$$
Does this solution look correct? I tried going through with the quotient rule, but it is a bit messy, so I was hoping someone could check my math
$$C'(\eta) = \frac{\left[3\left(1-\frac{\eta}{\eta_c}\right)\sum\limits_{k=1}^{3} B_kk\left(\frac{\eta}{\eta_c}\right)^{k-1}\frac{1}{\eta_c}\right]-\left[\left(1+\sum\limits_{k=1}^{3} B_k\left(\frac{\eta}{\eta_c}\right)^k\right)\left(-\frac{3}{\eta_c}\right)\right]}{\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^2}$$
 [/INST] Here's how you could differentiate this particular function:
$$
C'(\eta) = \left[\frac{1+\sum\limits_{k=1}^{3} B_k\left(\frac{\eta}{\eta_c}\right)^k}{3\left(1-\frac{\eta}{\eta_c}\right)}\right]'=
\left[\frac{1+B_{1}\frac{\eta}{\eta_c}+B_{2}\frac{\eta^2}{\eta_c^2}+B_{3}\frac{\eta^3}{\eta_c^3}}{3\left(1-\frac{\eta}{\eta_c}\right)}\right]'=\\
\left[\left(1+B_{1}\frac{\eta}{\eta_c}+B_{2}\frac{\eta^2}{\eta_c^2}+B_{3}\frac{\eta^3}{\eta_c^3}\right)\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^{-1}\right]'=\\
\left(1+B_{1}\frac{\eta}{\eta_c}+B_{2}\frac{\eta^2}{\eta_c^2}+B_{3}\frac{\eta^3}{\eta_c^3}\right)'\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^{-1}+\left(1+B_{1}\frac{\eta}{\eta_c}+B_{2}\frac{\eta^2}{\eta_c^2}+B_{3}\frac{\eta^3}{\eta_c^3}\right)\left(\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^{-1}\right)'=\\
\frac{B_{1}\frac{1}{\eta_c}+B_{2}\frac{2\eta}{\eta_c^2}+B_{3}\frac{3\eta^2}{\eta_c^3}}{3\left(1-\frac{\eta}{\eta_c}\right)}+\left(1+B_{1}\frac{\eta}{\eta_c}+B_{2}\frac{\eta^2}{\eta_c^2}+B_{3}\frac{\eta^3}{\eta_c^3}\right)(-1)\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^{-2}\left(3-3\frac{\eta}{\eta_c}\right)'=\\
\frac{\sum\limits_{k=1}^{3}B_k\frac{k\eta^{k-1}}{\eta_c^k}}{3\left(1-\frac{\eta}{\eta_c}\right)}+\frac{1+\sum\limits_{k=1}^{3} B_k\left(\frac{\eta}{\eta_c}\right)^k}{\left[3\left(1-\frac{\eta}{\eta_c}\right)\right]^2}\cdot\frac{3}{\eta_c}=\\
\frac{\sum\limits_{k=1}^{3}B_k\frac{k\eta^{k-1}}{\eta_c^k}}{3\left(1-\frac{\eta}{\eta_c}\right)}+\frac{1+\sum\limits_{k=1}^{3} B_k\left(\frac{\eta}{\eta_c}\right)^k}{3\eta_c\left(1-\frac{\eta}{\eta_c}\right)^2}.
$$
The answer I get appears to be equivalent to what you got.
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