1. The problem asks to differentiate functions using \(dy\) and explain the meaning or purpose of \(d\) in it. 2. In calculus, \(dy\) represents the differential of \(y\), which is the change in the function \(y\) corresponding to a small change in the variable \(x\). 3. The symbol \(d\) stands for "differential," indicating an infinitesimally small change in a variable. 4. When we write \(\frac{dy}{dx}\), it means the derivative of \(y\) with respect to \(x\), or the rate at which \(y\) changes as \(x\) changes. 5. To differentiate a function \(y=f(x)\), we find \(\frac{dy}{dx}\) by applying differentiation rules. 6. For example, if \(y = x^2\), then \(\frac{dy}{dx} = 2x\). 7. The differential \(dy\) can be expressed as \(dy = \frac{dy}{dx} dx\), meaning the change in \(y\) equals the derivative times the change in \(x\). 8. This concept helps approximate changes in functions and is fundamental in calculus. Final answer: \(d\) in \(dy\) denotes a very small change in \(y\), and differentiating functions using \(dy\) means finding how \(y\) changes with respect to \(x\).