Write down the definition of the derivative from first principles.

Here is the solution to the question. Question 1: Find from first principle the differential coefficient of the function y = 2x^2 with respect to x. Step 1: Write down the definition of the derivative from first principles. The derivative of a function f(x) from first principles is given by: (dy)/(dx) = _h 0 (f(x+h) - f(x))/(h) Given the function f(x) = 2x^2. First, we need to find f(x+h) by replacing x with (x+h) in the function: f(x+h) = 2(x+h)^2 Expand the term (x+h)^2: (x+h)^2 = x^2 + 2xh + h^2 Substitute this back into the expression for f(x+h): f(x+h) = 2(x^2 + 2xh + h^2) = 2x^2 + 4xh + 2h^2 Step 2: Calculate the difference f(x+h) - f(x). Subtract f(x) from f(x+h): f(x+h) - f(x) = (2x^2 + 4xh + 2h^2) - (2x^2) f(x+h) - f(x) = 4xh + 2h^2 Step 3: Divide the difference by h. Now, divide the result from Step 2 by h: (f(x+h) - f(x))/(h) = (4xh + 2h^2)/(h) Factor out h from the numerator: (h(4x + 2h))/(h) Cancel out h from the numerator and the denominator (since h ≠ 0 as h approaches 0): 4x + 2h Step 4: Take the limit as h 0. Finally, apply the limit as h approaches 0 to the simplified expression: (dy)/(dx) = _h 0 (4x + 2h) Substitute h=0 into the expression: (dy)/(dx) = 4x + 2(0) (dy)/(dx) = 4x The differential coefficient is 4x. 3 done, 2 left today. You're making progress.