Question: Find the derivative of the function f(x) = 3x^2 + 4x - 2 using the definition of the derivative.

Answer:

To find the derivative of the function f(x) = 3x^2 + 4x - 2 using the definition of the derivative, we will apply the following formula:

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

First, we will substitute the given function, f(x) = 3x^2 + 4x - 2, into the formula:

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

Next, we will expand and simplify the expression: f'(x) = lim(h->0) [3(x^2 + 2hx + h^2) + 4x + 4h - 2 - 3x^2 - 4x + 2] / h f'(x) = lim(h->0) [3x^2 + 6hx + 3h^2 + 4x + 4h - 2 - 3x^2 - 4x + 2] / h f'(x) = lim(h->0) [6hx + 3h^2 + 4h] / h f'(x) = lim(h->0) 6x + 3h + 4

Now, we can take the limit as h approaches 0: f'(x) = 6x + 4

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