Step 1: Start with the function f(x) = 4x^3 - 2x^2 + x.
Step 2: Differentiate f(x) to find the first derivative f'(x). Use the power rule: for each term, multiply by the exponent and decrease the exponent by 1.
Step 3: For the term 4x^3, the derivative is 12x^2 (4 * 3 = 12, and 3 - 1 = 2).
Step 4: For the term -2x^2, the derivative is -4x (-2 * 2 = -4, and 2 - 1 = 1).
Step 5: For the term x, the derivative is 1 (1 * 1 = 1, and 1 - 1 = 0).
Step 6: Combine the derivatives from Steps 3, 4, and 5 to get f'(x) = 12x^2 - 4x + 1.
Step 7: Now, differentiate f'(x) to find the second derivative f''(x). Again, use the power rule.
Step 8: For the term 12x^2, the derivative is 24x (12 * 2 = 24, and 2 - 1 = 1).
Step 9: For the term -4x, the derivative is -4 (the coefficient remains the same, and the exponent decreases to 0).
Step 10: The constant term 1 has a derivative of 0 (since the derivative of any constant is 0).
Step 11: Combine the derivatives from Steps 8, 9, and 10 to get f''(x) = 24x - 4.
Differentiation – The process of finding the derivative of a function, which measures how the function changes as its input changes.
Second Derivative – The derivative of the derivative, which provides information about the curvature of the function and can indicate concavity.
