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

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

• A. f'(x) = x
• B. f'(x) = 2x
• C. f'(x) = x²
• D. f'(x) = 2

Answer: (B)

The derivative of a function measures how the function's output changes as the input changes. For the function f(x) = x², we can apply the power rule of differentiation, which states that if f(x) = x^n, then the derivative f'(x) = n * x^(n-1). In this case, the exponent n is 2. According to the power rule: 1. Multiply the exponent (2) by the coefficient of the term (which is 1 in this case, since there is no coefficient written explicitly), resulting in 2. 2. Decrease the exponent by one: 2 - 1 = 1. Combining these steps gives us f'(x) = 2 * x^1, which simplifies to f'(x) = 2x. Hence, the correct derivative of the given function f(x) = x² is f'(x) = 2x. This result indicates that at any point x, the rate of change of the function f(x) is proportional to both the value of x and the constant 2. Understanding this derivative is important for applications in physics, economics, and many other fields where rates of change are analyzed.

The derivative of a function measures how the function's output changes as the input changes. For the function f(x) = x², we can apply the power rule of differentiation, which states that if f(x) = x^n, then the derivative f'(x) = n * x^(n-1).

In this case, the exponent n is 2. According to the power rule:

1. Multiply the exponent (2) by the coefficient of the term (which is 1 in this case, since there is no coefficient written explicitly), resulting in 2.

2. Decrease the exponent by one: 2 - 1 = 1.

Combining these steps gives us f'(x) = 2 * x^1, which simplifies to f'(x) = 2x.

Hence, the correct derivative of the given function f(x) = x² is f'(x) = 2x. This result indicates that at any point x, the rate of change of the function f(x) is proportional to both the value of x and the constant 2. Understanding this derivative is important for applications in physics, economics, and many other fields where rates of change are analyzed.