What is the derivative of x^2?

What is the derivative of x^2?

• A. 2x
• B. x^2
• C. 2
• D. x

Answer: (A)

The derivative of a function measures how the function's value changes as its input changes. For the function \( f(x) = x^2 \), we apply the power rule of differentiation, which states that when differentiating \( x^n \), the derivative is \( n \cdot x^{n-1} \). In this case, \( n = 2 \). Therefore, applying the power rule: 1. Multiply \( n \) (which is 2) by the variable raised to one less than the original power, which is \( x^{2-1} \) or \( x^1 \). 2. This results in \( 2 \cdot x^1 \), which simplifies to \( 2x \). Thus, the derivative of \( x^2 \) is \( 2x \), making that the correct answer. This reflects how the function's rate of change at any point \( x \) can be found, showing that the slope of the tangent line to the curve \( x^2 \) at any point is proportional to \( 2x \).

The derivative of a function measures how the function's value changes as its input changes. For the function ( f(x) = x^2 ), we apply the power rule of differentiation, which states that when differentiating ( x^n ), the derivative is ( n \cdot x^{n-1} ).

In this case, ( n = 2 ). Therefore, applying the power rule:

1. Multiply ( n ) (which is 2) by the variable raised to one less than the original power, which is ( x^{2-1} ) or ( x^1 ).

2. This results in ( 2 \cdot x^1 ), which simplifies to ( 2x ).

Thus, the derivative of ( x^2 ) is ( 2x ), making that the correct answer. This reflects how the function's rate of change at any point ( x ) can be found, showing that the slope of the tangent line to the curve ( x^2 ) at any point is proportional to ( 2x ).