differentiate f with respect to x, y, z, and w respectively. f_x = 2x f_y = 2y - 8w f_z = 2z f_w = - 8y+6w…

differentiate f with respect to x, y, z, and w respectively. f_x = 2x f_y = 2y - 8w f_z = 2z f_w = - 8y+6w step 3 apply the chain rule for implicit differentiation to find ∂w/∂x = - f_x/f_w

differentiate f with respect to x, y, z, and w respectively. f_x = 2x f_y = 2y - 8w f_z = 2z f_w = - 8y+6w step 3 apply the chain rule for implicit differentiation to find ∂w/∂x = - f_x/f_w

Answer

Explanation:

Step1: Recall implicit - differentiation formula

For an implicit function $F(x,y,z,w) = 0$, the formula for $\frac{\partial w}{\partial x}$ using the chain - rule for implicit differentiation is $\frac{\partial w}{\partial x}=-\frac{F_x}{F_w}$.

Step2: Identify $F_x$ and $F_w$

We are given that $F_x = 2x$ and $F_w=-8y + 6w$.

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

Substitute $F_x$ and $F_w$ into the formula: $\frac{\partial w}{\partial x}=-\frac{2x}{-8y + 6w}=\frac{2x}{8y - 6w}$.

Answer:

$\frac{2x}{8y - 6w}$