To find the derivative of the function f(x) = 2x² + 4x + 1, we will use the basic rules of differentiation.
The derivative of a function gives us the slope of the function at any point and can be found using the power rule. The power rule states that if you have a term in the form of ax^n, its derivative is n * ax^(n-1).
1. For the first term, 2x²: using the power rule, the derivative is 2 * 2x^(2-1) = 4x.
2. For the second term, 4x: this can be rewritten as 4x^1. The derivative is 1 * 4x^(1-1) = 4.
3. For the constant term 1: the derivative of a constant is 0.
Now, combine all these results:
f'(x) = 4x + 4 + 0.
Thus, the derivative of the function is:
f'(x) = 4x + 4.