Differentiate the following functions using the first principle: iii) y = 4x2 + 2x + 1 iv) y = sin x

Okay, let's differentiate these functions using the first principle. The definition of the derivative from first principles is: f'(x) = _h 0 (f(x+h) - f(x))/(h) iii) y = 4x^2 + 2x + 1 Step 1: Identify f(x) and f(x+h). f(x) = 4x^2 + 2x + 1 f(x+h) = 4(x+h)^2 + 2(x+h) + 1 f(x+h) = 4(x^2 + 2xh + h^2) + 2x + 2h + 1 f(x+h) = 4x^2 + 8xh + 4h^2 + 2x + 2h + 1 Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (4x^2 + 8xh + 4h^2 + 2x + 2h + 1) - (4x^2 + 2x + 1) f(x+h) - f(x) = 8xh + 4h^2 + 2h Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 (8xh + 4h^2 + 2h)/(h) f'(x) = _h 0 (h(8x + 4h + 2))/(h) f'(x) = _h 0 (8x + 4h + 2) Step 4: Evaluate the limit. f'(x) = 8x + 4(0) + 2 f'(x) = 8x + 2 iv) y = x Step 1: Identify f(x) and f(x+h). f(x) = x f(x+h) = (x+h) Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (x+h) - x Using the sum-to-product identity A - B = 2 ((A+B)/(2)) ((A-B)/(2)): f(x+h) - f(x) = 2 ((x+h+x)/(2)) ((x+h-x)/(2)) f(x+h) - f(x) = 2 (x + (h)/(2)) ((h)/(2)) Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 (2 (x + h)/(2)) ((h)/(2))h f'(x) = _h 0 [ (x + (h)/(2)) · ((h)/(2))(h)/(2) ] Step 4: Evaluate the limit using _ 0 ( )/() = 1. f'(x) = (x+0) · 1 f'(x) = x v) y = x Step 1: Identify f(x) and f(x+h). f(x) = x f(x+h) = (x+h) Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = (x+h) - x Using the sum-to-product identity A - B = -2 ((A+B)/(2)) ((A-B)/(2)): f(x+h) - f(x) = -2 ((x+h+x)/(2)) ((x+h-x)/(2)) f(x+h) - f(x) = -2 (x + (h)/(2)) ((h)/(2)) Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 (-2 (x + h)/(2)) ((h)/(2))h f'(x) = _h 0 [ -(x + (h)/(2)) · ((h)/(2))(h)/(2) ] Step 4: Evaluate the limit using _ 0 ( )/() = 1. f'(x) = -(x+0) · 1 f'(x) = - x vi) y = x Step 1: Identify f(x) and f(x+h). f(x) = x = ( x)/( x) f(x+h) = (x+h) = ((x+h))/((x+h)) Step 2: Calculate the difference f(x+h) - f(x). f(x+h) - f(x) = ((x+h))/((x+h)) - ( x)/( x) f(x+h) - f(x) = ((x+h) x - (x+h) x)/((x+h) x) Using the angle subtraction formula (A-B) = A B - A B: f(x+h) - f(x) = (((x+h)-x))/((x+h) x) = ( h)/((x+h) x) Step 3: Substitute into the limit definition and simplify. f'(x) = _h 0 ( h)/((x+h) x)h f'(x) = _h 0 [ ( h)/(h) · (1)/((x+h) x) ] Step 4: Evaluate the limit using _ 0 ( )/() = 1. f'(x) = 1 · (1)/((x+0) x) f'(x) = (1)/( x x) = (1)/(^2 x)