8) Find \( f^{\prime}(x) \) using the limit definition of the denivative Then evaluate at \( x=-2 \). \[ f(x)=-6 x^{2}-3 x+2 \]

To find f '(x) using the limit definition of the derivative, we recall that   f '(x) = limₕ→0 [(f(x + h) - f(x)) / h]. Given the function f(x) = -6x² - 3x + 2: 1. First, compute f(x + h):   f(x + h) = -6(x + h)² - 3(x + h) + 2.   Expand (x + h)²:    (x + h)² = x² + 2xh + h².   Thus:    f(x + h) = -6(x² + 2xh + h²) - 3x - 3h + 2          = -6x² - 12xh - 6h² - 3x - 3h + 2. 2. Find the difference f(x + h) - f(x):   f(x + h) - f(x) = [-6x² - 12xh - 6h² - 3x - 3h + 2] - (-6x² - 3x + 2)            = -12xh - 6h² - 3h. 3. Factor h from the numerator:   f(x + h) - f(x) = h(-12x - 6h - 3). 4. Divide by h to form the difference quotient:   [f(x + h) - f(x)]/h = -12x - 6h - 3. 5. Take the limit as h → 0:   f '(x) = limₕ→0 (-12x - 6h - 3)        = -12x - 3. Now, evaluate f '(x) at x = -2:   f '(-2) = -12(-2) - 3       = 24 - 3       = 21. Therefore, the derivative of the function is f '(x) = -12x - 3, and at x = -2, the derivative is 21.