1. The problem is to understand and use the notation \( \frac{dy}{dx} \), which represents the derivative of a function \( y \) with respect to \( x \).\n\n2. The derivative \( \frac{dy}{dx} \) measures how the function \( y \) changes as \( x \) changes. It is the limit of the average rate of change as the change in \( x \) approaches zero.\n\n3. The formula for the derivative of a function \( y = f(x) \) is given by:\n$$\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$\nThis means we look at the ratio of the change in \( y \) to the change in \( x \) as the change in \( x \) becomes very small.\n\n4. Important rules for derivatives include:\n- The power rule: \( \frac{d}{dx} x^n = n x^{n-1} \)\n- The sum rule: \( \frac{d}{dx} (f(x) + g(x)) = \frac{df}{dx} + \frac{dg}{dx} \)\n- The constant multiple rule: \( \frac{d}{dx} [c f(x)] = c \frac{df}{dx} \) where \( c \) is a constant.\n\n5. Example: If \( y = x^3 + 2x \), then using the power rule and sum rule:\n$$\frac{dy}{dx} = 3x^2 + 2$$\nThis means the rate of change of \( y \) with respect to \( x \) is \( 3x^2 + 2 \).\n\n6. In summary, \( \frac{dy}{dx} \) is a fundamental concept in calculus that tells us how a function changes at any point \( x \). It is used extensively in physics, engineering, and other sciences to analyze rates of change.