Step 1 To find the derivative of the function, we will use the formula f'(a) = lim f(x) = f(a) xa x-a For f(x)=x, we have f'(a) = lim xa f'(a) = We can factor x a as the difference of two cubes, or x - a = (x/3 a/3)(x/3 + x/3a/3 + a/3). Substituting this back into the limit and canceling common factors gives us x1/3 - 1/3 (x1/3 a1/3)(x2/3 + x1/3a1/3 + a/3) = lim xa x1/3 a1/3 = lim xa f'(a) = lim x-a Step 2 We can now evaluate this limit to conclude that the derivative is as follows. 1 (x2/3 + x1/3 a1/3 + a2/3) 1 xa (x2/3 + x/3 a/3 + a2/3) a (3) 1 Exercise (b) Show that f '(0) does not exist. Step 1 Using f '(x) = lim f(x + h) f(x) with x = 0, we have h0 h Submit f'(0) = lim f(0+h)-f(0) h0 h = lim h0 = lim h0 /0+h h 1 -0 Skip (you cannot come back) X