What is the second derivative of y = 4x^2?

The second derivative of \( y = 4x^2 \) is 8.

To find the second derivative, we first need to find the first derivative of the function \( y = 4x^2 \). The first derivative, often denoted as \( \frac{dy}{dx} \) or \( y' \), represents the rate of change of \( y \) with respect to \( x \). Using the power rule for differentiation, which states that \( \frac{d}{dx} [x^n] = nx^{n-1} \), we can differentiate \( 4x^2 \).

Applying the power rule:
\[ \frac{d}{dx} [4x^2] = 4 \cdot 2x^{2-1} = 8x \]

So, the first derivative of \( y = 4x^2 \) is \( 8x \).

Next, we need to find the second derivative, which is the derivative of the first derivative. The second derivative, denoted as \( \frac{d^2y}{dx^2} \) or \( y'' \), tells us how the rate of change itself is changing. We differentiate \( 8x \) using the power rule again.

Applying the power rule to \( 8x \):
\[ \frac{d}{dx} [8x] = 8 \cdot 1x^{1-1} = 8 \]

Therefore, the second derivative of \( y = 4x^2 \) is 8. This means that the rate of change of the slope of the curve \( y = 4x^2 \) is constant and equal to 8.