In this section, we focus on the applications of the derivative. Despite the fact that the definition of the derivative is rather abstract (using the limit of the ratio of the increments of the function and the independent variable), the fields of its applications are extremely diverse. With the help of the derivative, one can solve such problems as investigation of functions and sketching their graphs, optimization of various systems and modes of operations, simplifying algebraic expressions, approximate calculations, and much more. The topics given below cover the main applications of the derivative and contain a lot of practical problems with detailed solutions. All the material is intended for high school and college students.

• Tangent and Normal Lines
• Rolle’s Theorem
• Lagrange’s Mean Value Theorem
• Cauchy’s Mean Value Theorem
• Newton’s Method
• Related Rates
• Linear Approximation
• Rectilinear Motion
• Planar Motion
• Critical Points
• Increasing and Decreasing Functions
• Local Extrema of Functions
• Global Extrema of Functions
• Convex Functions
• Inflection Points
• Asymptotes
• Curve Sketching
• Classical Inequalities
• Proving of Inequalities
• Curvature and Radius of Curvature
• Evolute and Involute
• Envelope of a Family of Curves
• Osculating Curves
• Optimization Problems in 2D Geometry
• Optimization Problems in 3D Geometry
• Optimization Problems Involving Numbers
• Optimization Problems in Physics
• Optimization Problems in Economics
• Van der Waals Equation