Question

step 4 apply the chain rule for implicit differentiation to find $\frac{partial w}{partial y}=-\frac{f_y}{f_w}$ and $\frac{partial w}{partial x}=-\frac{f_x}{f_w}$. $\frac{2x}{8y - 6w}$

Explanation:

Step1: Recall implicit - differentiation chain - rule formula

For an implicit function $F(x,y,w) = 0$, the formula for $\frac{\partial w}{\partial x}=-\frac{F_x}{F_w}$ and $\frac{\partial w}{\partial y}=-\frac{F_y}{F_w}$.

Step2: Identify $F_x,F_y,F_w$

Assume $F(x,y,w)$ is such that $F_x = 2x$, $F_y=8y - 6w$.

Step3: Calculate $\frac{\partial w}{\partial x}$

$\frac{\partial w}{\partial x}=-\frac{F_x}{F_w}=-\frac{2x}{F_w}$ (not given in full in the problem - statement for $F_w$).

Step4: Calculate $\frac{\partial w}{\partial y}$

$\frac{\partial w}{\partial y}=-\frac{F_y}{F_w}=-\frac{8y - 6w}{F_w}$

Answer:

$\frac{\partial w}{\partial y}=-\frac{8y - 6w}{F_w}$