Step 1 To find the derivative of the function, we will use the formula f'(a) =...

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Step 1 To find the derivative of the function, we will use the formula f'(a) = lim f(x) = f(a) x→a x-a For f(x)=√x, we have f'(a) = lim x→a f'(a) = We can factor x a as the difference of two cubes, or x - a = (x¹/3 a¹/3)(x²/3 + x¹/3a¹/3 + a²/3). 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 + a²/3) = lim x→a x1/3 a1/3 = lim x→a 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 x→a (x2/3 + x¹/3 a¹/3 + a2/3) a (3²) 1 Заз 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 h→0 h Submit f'(0) = lim f(0+h)-f(0) h→0 h = lim h→0 = lim h→0 ¾/0+h h 1 -0 Skip (you cannot come back) X Step 1 To find the derivative of the function, we will use the formula f'(a) = lim f(x) = f(a) x→a x-a For f(x)=√x, we have f'(a) = lim x→a f'(a) = We can factor x a as the difference of two cubes, or x - a = (x¹/3 a¹/3)(x²/3 + x¹/3a¹/3 + a²/3). 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 + a²/3) = lim x→a x1/3 a1/3 = lim x→a 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 x→a (x2/3 + x¹/3 a¹/3 + a2/3) a (3²) 1 Заз 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 h→0 h Submit f'(0) = lim f(0+h)-f(0) h→0 h = lim h→0 = lim h→0 ¾/0+h h 1 -0 Skip (you cannot come back) X