Answer:
Step-by-step explanation:Let's find the first derivative of f(x) using the first principle. Here is the formula for finding the first derivative:f'(x) = [f(x + h) - f(x)] / h
Where h is a small number that represents an infinitesimal change in x.
Now let's apply this formula to the given function f(x) = -2x² + bx + 5.
f'(x) = [f(x + h) - f(x)] / h
f'(x) = [-2(x + h)² + b(x + h) + 5 - (-2x² + bx + 5)] / h
f'(x) = [-2(x² + 2xh + h²) + bx + bh + 5 + 2x² - bx - 5] / h
f'(x) = [-2x² - 4xh - 2h² + bx + bh] / h
f'(x) = (-2x² + bx + 4xh + 2h² + bh) / h
Taking the limit as h approaches 0 gives us the derivative at x:
f'(x) = lim(h→0) (-2x² + bx + 4xh + 2h² + bh) / h
f'(x) = -4x + b
Therefore, the first derivative of f(x) is f'(x) = -4x + b.