To find the derivative of the function f(x) = 2x² + 4x + 1, we will use basic rules of differentiation. The derivative, denoted as f'(x), measures how the function changes as x changes.
We’ll differentiate each term of the function separately:
- The term 2x²: Using the power rule, which states that the derivative of x^n is n*x^(n-1), we find that:
2 * 2x^{2-1} = 4x
- The term 4x: For a linear term like this, the derivative is simply the coefficient of x:
4
- The constant term 1: The derivative of any constant is
0.
Now, we can combine all these results to find the derivative of f(x):
f'(x) = 4x + 4 + 0
So the final answer is:
f'(x) = 4x + 4