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I have a function $$Y_a= [\alpha_2(\alpha_1 K^{\frac{\sigma_1-1}{\sigma_1}}+(1-\alpha_1)AR^{\frac{\sigma_1-1}{\sigma_1}})^{\frac{\sigma_1}{\sigma_1-1}\frac{\sigma_2-1}{\sigma_2}}+(1-\alpha_2)L^{\frac{\sigma_2-1}{\sigma_2}}]^{\frac{\sigma_2}{\sigma_2-1}} $$ And I need to take a first order derivative w.r.t $K$ . It becomes $$ K=\frac{r}{z^{\frac{1}{\sigma_1-1}} (\sigma_1 L_1)^{\frac{\sigma_1-1}{\sigma_1}}}, $$ where $$ z=(1-\alpha_1)A_1*R)^{\frac{\sigma_1-1}{\sigma_1}}+(\alpha_1*K_1*L_1)^{\frac{\sigma_1-1}{\sigma_1}} $$ Then this equation is an implicit function of $K$ and $A$, taken everything else as parameters.

My question is how to get a derivative of $K$ w.r.t $A$ and I need to know the sign of this derivative. $\alpha$ is between $0$ and $1$, $\sigma$ can be larger or smaller than $1$, $R$ and $L$ are positive.

I tried matlab symbolic operations but it says too many output arguments.

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