Differentiate the following using the first principle
I. f (x)2-2x
To differentiate the function f(x) = x^2 - 2x using the first principle, we need to apply the definition of the derivative:
f'(x) = lim (h -> 0) [f(x + h) - f(x)] / h
Plugging in our function f(x) = x^2 - 2x:
f'(x) = lim (h -> 0) [(x + h)^2 - 2(x + h) - (x^2 - 2x)] / h
f'(x) = lim (h -> 0) [x^2 + 2hx + h^2 - 2x - 2h - x^2 + 2x] / h
f'(x) = lim (h -> 0) [2hx + h^2 - 2h] / h
Now, we can simplify by factoring out an h from the numerator:
f'(x) = lim (h -> 0) [h(2x + h - 2)] / h
Finally, we can cancel out the h in the numerator and denominator:
f'(x) = lim (h -> 0) 2x + h - 2
f'(x) = 2x - 2
Therefore, the derivative of f(x) = x^2 - 2x with respect to x using the first principle is f'(x) = 2x - 2.