Final answer:
The value of gs(4, 2) = 5 * 2 = 10.
gr(4, 2) = 30 and gs(4, 2) = 10.
Step-by-step explanation:
To calculate gr(4, 2) and gs(4, 2) using the table of values provided, we need to use the chain rule of differentiation.
Let's start with gr(4, 2):
gr(4, 2) represents the partial derivative of g with respect to r at the point (4, 2).
Using the chain rule, we have:
gr(4, 2) = (d/dx) [f(5x - y,- 7x)] * (d/dx) [5r - s]
The first part, (d/dx) [f(5x - y, - 7x)], represents the partial derivative of f with respect to x, evaluated at (5r - s, - 7r).
Looking at the given table, we can see that fx = 6 at the point (4, 2). Therefore, (d/dx) [f(5x - y, - 7x)] = 6.
The second part, (d/dx) [5r - s], represents the partial derivative of 5r - s with respect to r.
Taking the derivative with respect to r, we get:
(d/dx) [5r - s] = 5.
Therefore, gr(4, 2) = 6 * 5 = 30.
Now, let's calculate gs(4, 2):
gs(4, 2) represents the partial derivative of g with respect to s at the point (4, 2).
Using the chain rule, we have:
gs(4, 2) = (d/dx) [f(5x - y, - 7x)] * (d/dx) [ - 7r]
Using the given table, fy = 5 at the point (4, 2). Therefore, (d/dx) [f(5x - y, - 7x)] = 5.
Taking the derivative of - 7r with respect to s, we get:
(d/dx) - 7r] = 2s.