Derivatives

Derivatives measure how a function changes as its input changes, capturing the concept of instantaneous rate of change. This topic covers differentiation rules from the basic power rule through advanced techniques like implicit differentiation and logarithmic differentiation, along with applications such as related rates and optimization.

Derivatives are essential in physics for describing motion and forces, in economics for marginal analysis, and in engineering for optimizing designs. Any field that models change relies on differentiation.

Practice Tips

• 1Master the power, product, and chain rules before tackling implicit or logarithmic differentiation.
• 2For related rates problems, draw a diagram and identify which quantities change with time before writing equations.
• 3When optimizing, always verify your critical point is a maximum or minimum using the second derivative test.