I doing some multivariable calculus work and was struggling with the following question.
Given: $$ e^{7z} = xyz $$ The task is to compute the partial derivatives dz/dx and dz/dy using implicit differentiation. My solution is now as follows: $$ (dz/dx) 7e^{7z} = yz + (dz/dx)xy $$ And so, $$ (dz/dx) (7e^{7z}-xy) = yz $$ Therefore correct answer is, $$ \frac{dz}{dx} = \frac{yz}{(7e^{7z}-xy)} $$
Since there are three variables so it will be differentiated by product rule i.e. $\frac{dz}{dx}7e^{7z}= yz+ \frac{dy}{dx}xz+ \frac{dz}{dx}xy$