Final answer:
To find the derivative of the function f(x) = ln |x^3 - 3x + 2|, we can use the chain rule. We find f'(x) = { 3x^2 - 3, x >= 0, -3x^2 + 3, x < 0 }. Using this, we can calculate f'(0) = -3 and f'(2) = 6.
Step-by-step explanation:
To find the derivative of the function f(x) = ln |x^3 - 3x + 2|, we can use the chain rule. The chain rule states that if we have a composite function g(f(x)), the derivative is g'(f(x)) * f'(x). For our function, the outer function is the natural logarithm and the inner function is |x^3 - 3x + 2|. Let's find f'(x) first:
f'(x) = derivative of |x^3 - 3x + 2|
To take the derivative of |x^3 - 3x + 2|, we need to consider the different cases for x. We can split the function into two cases: x^3 - 3x + 2 when x >= 0 and -(x^3 - 3x + 2) when x < 0. Taking the derivative of these two cases separately, we get:
f'(x) = { 3x^2 - 3, x >= 0, -3x^2 + 3, x < 0 }
Now that we have f'(x), we can find f'(0) and f'(2) by plugging in the respective values of x:
f'(0) = 3(0)^2 - 3 = -3
f'(2) = 3(2)^2 - 3 = 9 - 3 = 6