Final answer:
To find the total derivative of g∘f at a specified point a, we need to apply the Chain Rule. The steps involve substituting the point a into f and then into g, finding the partial derivatives of g and f(a), and multiplying them together using the Chain Rule.
Step-by-step explanation:
To find the total derivative of g∘f at a specified point a, we need to apply the Chain Rule. The Chain Rule states that if we have two functions, g and f, and we want to find the derivative of the composite function g∘f, then the derivative is given by the product of the derivative of g with respect to its variables and the derivative of f with respect to its variables.
In this case, g(x,y) = (xy², x²ʸ, x³+y^3) and f(x,y) = (x−y, x²-y²). We are asked to find the total derivative of g∘f at the point a. The point a is given by f(a) = (a−b, a²-b²), so we can substitute the expression for f(a) into g to get g(f(a)). After substituting, we can apply the Chain Rule to find the total derivative.
The steps are as follows:
- Find f(a) by substituting a into f(x,y).
- Substitute f(a) into g(x, y) to get g(f(a)).
- Find the partial derivatives of g with respect to x and y.
- Find the partial derivatives of f(a) with respect to x and y.
- Apply the Chain Rule by multiplying the partial derivatives of g and f(a) together.
- Simplify the result.