Answer:
dz = (6x^2y^9)dx + (18x^3y^8)dy.
Explanation:
To find the total differential of the function z = 2x^3y^9, we need to compute its partial derivatives with respect to x and y, and then express the total differential in terms of dx and dy.
Let's start by finding the partial derivative of z with respect to x, assuming y is constant:
∂z/∂x = ∂(2x^3y^9)/∂x
= 6x^2y^9
Next, let's find the partial derivative of z with respect to y, assuming x is constant:
∂z/∂y = ∂(2x^3y^9)/∂y
= 18x^3y^8
Now that we have the partial derivatives, we can express the total differential of z as:
dz = (∂z/∂x)dx + (∂z/∂y)dy
Substituting the partial derivatives we found earlier:
dz = (6x^2y^9)dx + (18x^3y^8)dy
So, the total differential of z = 2x^3y^9 is given by dz = (6x^2y^9)dx + (18x^3y^8)dy.