Question:

Find the derivative of the function f(x) = 2x^3 - 5x^2 + 4x - 3 using first principles.

Answer:

To find the derivative of the given function using first principles, we start by using the definition of the derivative:

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

Substitute the function f(x) = 2x^3 - 5x^2 + 4x - 3 into the definition:

f'(x) = lim (h -> 0) [(2(x + h)^3 - 5(x + h)^2 + 4(x + h) - 3) - (2x^3 - 5x^2 + 4x - 3)] / h

Expand the expressions within the limit:

f'(x) = lim (h -> 0) [(2(x^3 + 3x^2h + 3xh^2 + h^3) - 5(x^2 + 2xh + h^2) + 4x + 4h - 3) - (2x^3 - 5x^2 + 4x - 3)] / h

f'(x) = lim (h -> 0) [2x^3 + 6x^2h + 6xh^2 + 2h^3 - 5x^2 - 10xh - 5h^2 + 4x + 4h - 3 - 2x^3 + 5x^2 - 4x + 3] / h

Combine like terms and simplify:

f'(x) = lim (h -> 0) [6x^2h + 6xh^2 + 2h^3 - 10xh - 5h^2 + 4h] / h

Factor out the common terms:

f'(x) = lim (h -> 0) h[6x^2 + 6xh + 2h^2 - 10x - 5h + 4] / h

Cancel out the h:

f'(x) = 6x^2 + 4 - 10x

Therefore, the derivative of the function f(x) = 2x^3 - 5x^2 + 4x - 3 is f'(x) = 6x^2 + 4 - 10x.