Derivative of x?? Use the symbolic capabilities of a calculator to calculate f?(x) f(x+h)?f(x) using the definition lim h for the following functions. h?0 a.? f(x) = x2 b.? f(x) = x3 c?. f(x) = x4 d.? f(x) = x100 e.? Based upon your answers to parts (a)–(d), propose a formula for f?(x) if f(x) = x? where n is a positive integer.

SOLUTION f(x+h)f(x) the definition of derivative f(x) = lim h h0 STEP 1 2 (a).Consider the function f(x) = x By using MAPLE,we can calculate the derivative by using the above formula (x+h) x The input command will be lim h h0 Then we will get the output as 2x. Thus, the derivative of the function f(x) = x is f(x) = 2x. We can check the above derivative by manual calculations 2 2 f(x) = limf(x+h)f(x= lim (x+h) x h0 h h0 h 2 = lim x +h +2xhx = lim h(h+2x) = limh + 2x = 2x h0 h h0 h h0 Thus we got the derivative of the function f(x) = x as f(x) = 2x STEP 2 (b).Consider the function f(x) = x 3 By using MAPLE,we can calculate the derivative by using the above formula 3 3 The input command will be lim (x+h) x h0 h Then we will get the output as 3x . 2 Thus, the derivative of the function f(x) = x is f(x) = 3x . 2 We can check the above derivative by manual calculations f(x+h)f(x) (x+h) x3 f(x) = lim = lim h0 h h0 h x +h +3x h+3xh x3 h(h +3x +3xh) 2 = lim h = lim h = limh + 3x + 3xh = 3x 2 h0 h0 h0 Thus we got the derivative of the function f(x) = x as f(x) = 3x 2 STEP 3 4 (c).Consider the function f(x) = x By using MAPLE,we can calculate the derivative by using the above formula (x+h) x The input command will be lim h h0 Then we will get the output as 4x . 3 Thus, the derivative...