Final answer:
To find the partial derivatives ∂z/∂x and ∂z/∂y, implicit differentiation is used on the given equation, applying the product rule and the chain rule where necessary, and solving for the respective derivatives.
Step-by-step explanation:
The question from the student involves finding the partial derivatives ∂z/∂x and ∂z/∂y of the given implicit function yz5 + xz4 + 5z2 + ln(9x + y) = 0 by using implicit differentiation. Since the function is not explicitly solved for z, we differentiate both sides of the equation with respect to x and y, treating z as a function of both x and y (z = z(x,y)). As we differentiate term by term, we apply the product rule and the chain rule accordingly.
For example, differentiating yz5 with respect to x, we treat y as a constant and get y∙(5z4)∙(∂z/∂x). Similarly, when differentiating xz4 with respect to x, we apply the product rule to obtain z4 + x∙(4z3)∙(∂z/∂x). We continue this process for each term and solve the resulting equations to find the desired partial derivatives.
To find ∂z/∂x and ∂z/∂y using implicit differentiation, we start by differentiating both sides of the equation with respect to x. We treat y and z as functions of x and apply the chain rule when differentiation terms involving y and z. Then, we differentiate both sides of the equation with respect to y, treating x and z as functions of y. Finally, we solve the resulting equations for ∂z/∂x and ∂z/∂y.