The value of _ x 0 (1-x)^n-1 x =

Step-by-step solution

Let f(x) = (1-x)^n . The given limit is of the form _ x 0 f(x) - f(0) x - 0 , which is the definition of the derivative f'(0) . Calculate the derivative of f(x) = (1-x)^n with respect to x : f'(x) = n(1-x)^ n-1 d dx (1-x) = n(1-x)^ n-1 (-1) = -n(1-x)^ n-1 Evaluate the derivative at x = 0 : f'(0) = -n(1-0)^ n-1 = -n(1)^ n-1 = -n . Answer: -n