What is the derivative of y = x^2?

The derivative of \( y = x^2 \) is \( \frac{dy}{dx} = 2x \).

To understand why this is the case, let's break it down. The derivative of a function tells us the rate at which the function's value changes as the input changes. In simpler terms, it gives us the slope of the tangent line to the curve at any given point.

For the function \( y = x^2 \), we use the power rule to find the derivative. The power rule states that if you have a function of the form \( y = x^n \), where \( n \) is a constant, the derivative is \( \frac{dy}{dx} = nx^{n-1} \).

In our case, \( y = x^2 \), so \( n = 2 \). Applying the power rule:
\[ \frac{dy}{dx} = 2x^{2-1} = 2x \]

This means that for any value of \( x \), the slope of the tangent line to the curve \( y = x^2 \) is \( 2x \). For example, if \( x = 1 \), the slope is \( 2 \times 1 = 2 \). If \( x = 3 \), the slope is \( 2 \times 3 = 6 \).

Understanding derivatives is crucial in many areas of mathematics and science because they help us analyse how things change. In this case, knowing that the derivative of \( y = x^2 \) is \( 2x \) allows us to predict how the function behaves as \( x \) increases or decreases.