Final Answers
(a) dv/dx at x=-1 is 0.
(b) du/dv at v=-1 is -6.
Step-by-step explanation
To determine the derivative dv/dx at x=-1, we begin by using the chain rule in differentiation. We first find dv/dw and then dw/dx. Substituting the given expressions, we find dv/dw = -2(w²-1)⁻³/² * 3 = -6/(w²-1)³/². Then, dw/dx = 3, as the derivative of 3x-1 with respect to x is simply 3. Finally, applying the chain rule, dv/dx = dv/dw * dw/dx = -6/(w²-1)³/² * 3 = -18/(w²-1)³/². Substituting w = 3x-1 and evaluating at x=-1 yields dv/dx at x=-1 as 0.
To find du/dv at v=-1, we employ the chain rule again. We start by determining du/dw and then dw/dv. Given
the derivative du/dv = -1/(2√(1-v)) by applying the chain rule. Next, substituting v = 2/(w²-1), the derivative dw/dv = -4/(w²-1)². Multiplying these derivatives together, du/dv * dw/dv = -1/(2√(1-v)) * -4/(w²-1)². Substituting v = -1 and w = 3x-1, we get du/dv at v=-1 as -6.