To find the derivative of the function f(x) = 4x^2 - 5x + 1, we should apply the power rule and addition/subtraction for derivatives. The power rule states that if you have a function of the form f(x) = x^n, then its derivative is given by f'(x) = n*x^(n-1). Moreover, the derivative of a constant is zero.
Step 1:
Let's start by finding the derivative of the first term, 4x^2. With the application of the power rule, its derivative would be 2 * 2 * x^(2-1) = 4x. But remember there's a 4 multiplying x^2, so the derivative becomes 4 * 4x = 16x.
Step 2:
Next, find the derivative of the second term -5x. Using the power rule, its derivative would be 1 * x^(1-1) = x^0 = 1. Multiplying this by the coefficient (-5) gives us -5.
Step 3:
The last term is a constant (1), and the derivative of a constant is zero, so it drops out.
Step 4:
Finally, combine the derivatives of the individual terms to get the derivative of the entire function, f'(x) = 8x - 5.