Finding the Derivative of a Function

Find the derivative of $y$ with respect to $x$.

Finding the derivative of a function is a fundamental concept in calculus, particularly for understanding rates of change and slopes of curves. When tasked with finding the derivative of y with respect to x, it is important to recognize this as an exercise in applying differentiation rules. You'll need to determine whether to use basic rules, such as the power rule, product rule, or the chain rule, depending on the form of the function y. Each rule provides a systematic way of handling different types of functions and combinations of functions, which is crucial in deriving correct solutions and understanding function behaviors.

In more advanced contexts, derivatives are not just about finding the slope at a point but also about modeling real-world phenomena. For instance, derivatives can describe how physical quantities change over time or space, making them essential in fields such as physics, engineering, and economics. In application, understanding the derivative can help interpret graphs, optimize functions, and solve equations that describe dynamic systems. This problem, while seemingly simple, lays the groundwork for tackling more complex problems involving derivative applications, such as optimization problems and analyzing motion.