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

The function ( f(x) = x^2 ) is a simple polynomial function, and to find its derivative, we can apply the power rule for differentiation. The power rule states that if you have a term ( x^n ), the derivative is given by ( n \times x^{n-1} ).

In this case, we have:

1. Identify ( n ) in the term ( x^2 ), which is 2.

2. According to the power rule, multiply this exponent by the coefficient (in this case, the coefficient is 1, which does not change the value) and then subtract 1 from the exponent.

Applying the rule:

• The derivative of ( x^2 ) becomes ( 2 \times x^{2-1} = 2x^1 = 2x ).

Therefore, the derivative ( f'(x) ) is ( 2x ), confirming that the correct answer is indeed represented accurately. This fundamental rule of calculus is crucial for understanding how to find the rate of change of polynomial functions, laying the groundwork for more complex differentiation scenarios.