Question

Suppose f is a differentiable function of x and y, and the following applies: p(t) = (g(t), h(t)) where g'(3) = -5, h(3) = 7, h'(3) = 2, f_x(6, 7) = 8, f_y(6, 7) = 2. Find p'(3), p(3).

a) p'(3) = (-5, 2), p(3) = (3, 7)
b) p'(3) = (-5, 2), p(3) = (6, 7)
c) p'(3) = (8, 2), p(3) = (3, 7)
d) p'(3) = (8, 2), p(3) = (6, 7)

Answer 1

Using the given derivatives of g and h at t=3, p'(3) can be found as (-5, 2). However, without the value of g(3), p(3) cannot be determined definitively from the given information. The correct answer is assumed to be provided, resulting in choosing one of the given options.

Step-by-step explanation:

The given problem involves the use of the chain rule for differentiation to find the derivative of the composition of functions, p(t), which is defined in terms of two other functions, g(t) and h(t). Since we are given the derivative of g at t=3 and the value and derivative of h at t=3, we can directly find p'(3) by pairing these values as (g'(3), h'(3)). To find p(3), we would typically need the values of g(3) and h(3). However, with the context provided, we are not given g(3) and hence cannot find p(3) with certainty from the details provided. Therefore, we can express p'(3) as (-5, 2), but we cannot conclude what p(3) will be from the given information. Despite this, it is assumed that the information given is sufficient and the provided answers are correct by the question context. Thus, we choose the option with the correct derivative from the given choices.