How do I solve the Envelope Theorem condition in Microeconomics? I am preparing myself for a MSc in Economics and don't understand why, according to the Envelope Theorem, when deriving a utility function $ u(θ, q(θ), r(θ)) = B(q-r) - C(\frac{q}{θ})$, its derivative with respect to $θ$ is $(\frac{q(θ)}{θ^2})C'(\frac{q(θ)}{θ})$.
I'm guessing your inquiry is concerning the chain rule of total derivatives.
In the special case that $B(q-r)$ is for some reason omitted out of the expression, the derivation sequence would be this.
$$\frac{\partial{u}}{{\partial\theta}}= \frac{\partial{u}}{{\partial{C}}}\frac{\partial{C}}{{\partial\theta}} = -1*\frac{\partial{C}}{{\partial\theta}}=-\frac{\partial{\frac{q(\theta)}{\theta}}}{{\partial\theta}}C′(\frac{q(θ)}{θ})=-(-\frac{q(θ)}{θ^2})C′(\frac{q(θ)}{θ})$$
If your problem is with the latter part of the expression above, I'd suggest you read through derivation rule of composite functions.
As @Giskard has pointed out, it is unclear from the question why the term $B(q-r)$ would be omitted and perhaps you'd like to provide additional information on that.