Question

Use implicit differentiation to find ∂z/∂x and ∂z/∂y. Given e⁹z = xyz, find ∂z/∂x and ∂z/∂y.

Answer 1

To find the partial derivatives ∂z/∂x and ∂z/∂y, differentiate the equation e9z = xyz implicitly with respect to x and y, and solve for dz/dx and dz/dy respectively.

Step-by-step explanation:

To find the partial derivatives ∂z/∂x and ∂z/∂y using implicit differentiation, we start with the given equation e9z = xyz. Differentiating both sides concerning x, we get:

9e9zdz/dx = yz + xyz(dz/dx)

Solving for dz/dx we have:

dz/dx = yz/(9e9z - xyz)

Similarly, differentiating both sides concerning y, we get:

9e9zdz/dy = xz + xyz(dz/dy)

Solving for dz/dy we have:

dz/dy = xz/(9e9z - xyz)

These expressions give us the partial derivatives of z concerning x and y respectively.