Find the second derivative of f(x) = 4x^4 - 2x^3 + x. (2019)

Find the second derivative of f(x) = 4x^4 - 2x^3 + x. (2019)

Step 1: Start with the function f(x) = 4x^4 - 2x^3 + x.
Step 2: To find the first derivative f'(x), apply the power rule to each term: for 4x^4, the derivative is 4 * 4x^(4-1) = 16x^3; for -2x^3, the derivative is -2 * 3x^(3-1) = -6x^2; for x, the derivative is 1.
Step 3: Combine the derivatives from Step 2 to get the first derivative: f'(x) = 16x^3 - 6x^2 + 1.
Step 4: Now, find the second derivative f''(x) by differentiating f'(x). Apply the power rule again: for 16x^3, the derivative is 16 * 3x^(3-1) = 48x^2; for -6x^2, the derivative is -6 * 2x^(2-1) = -12x; for the constant 1, the derivative is 0.
Step 5: Combine the derivatives from Step 4 to get the second derivative: f''(x) = 48x^2 - 12x.

Differentiation – The process of finding the derivative of a function, which measures the rate of change of the function.
Higher-order derivatives – The second derivative is the derivative of the first derivative, providing information about the curvature of the function.