What is the derivative of the function f(x) = 2x²?

To determine the derivative of the function ( f(x) = 2x^2 ), you apply the power rule of differentiation. The power rule states that if you have a function in the form of ( ax^n ), the derivative is given by ( nax^{n-1} ).

In this case, the function can be identified as ( f(x) = 2x^2 ), where ( a = 2 ) and ( n = 2 ). Applying the power rule:

1. Multiply the exponent (2) by the coefficient (2), which gives ( 2 \times 2 = 4 ).

2. Then subtract 1 from the exponent (2), yielding ( 2 - 1 = 1 ).

So, the derivative is ( f'(x) = 4x^{1} ), which simplifies to ( f'(x) = 4x ).

This derivation shows that the correct answer is indeed the one that states ( f'(x) = 4x ), clearly demonstrating the application of the power rule to find the derivative of the function.