-x^3 + x^2 + x first principles

Answer 1

To find the derivative of the function f(x) = -x^3 + x^2 + x using the first principles, we need to use the definition of the derivative:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

First, let's find f(x + h):

f(x + h) = -(x + h)^3 + (x + h)^2 + x + h
= -(x^3 + 3x^2h + 3xh^2 + h^3) + (x^2 + 2xh + h^2) + x + h
= -x^3 - 3x^2h - 3xh^2 - h^3 + x^2 + 2xh + h^2 + x + h

Next, let's substitute f(x + h) and f(x) into the definition of the derivative:

f'(x) = lim(h->0) [(-x^3 - 3x^2h - 3xh^2 - h^3 + x^2 + 2xh + h^2 + x + h) - (-x^3 + x^2 + x)] / h
= lim(h->0) [(-3x^2h - 3xh^2 - h^3 + 2xh + h^2 + h)] / h
= lim(h->0) [-3x^2 - 3xh - h^2 + 2x + h + 1]
= -3x^2 + 2x + 1

Therefore, the derivative of the function f(x) = -x^3 + x^2 + x using the first principles is f'(x) = -3x^2 + 2x + 1.