Applications of the derivative

In this section, we focus on the applications of the derivative. Although the definition of the derivative is rather abstract (in terms of the limit of the ratio of the function increment to the argument increment), its applications are extremely diverse. The derivative can be used to solve problems such as investigating and plotting functions, optimizing various systems and operating modes, simplifying algebraic expressions, performing approximate calculations, and much more. The topics discussed below cover the main applications of the derivative and include many practical problems with detailed solutions. All material is accessible to schoolchildren and students.

• • Differentials in approximate calculations
  • Monotonicity of functions
  • Local extrema of a function
  • The largest and smallest values ​​of a function
  • Convexity of functions
  • Inflection points
  • Asymptotes
  • Plotting function graphs
  • Rolle’s theorem
  • Lagrange’s theorem
  • Cauchy’s theorem
  • Classical inequalities
  • Proof of inequalities
  • Equations of tangent and normal
  • Curvature and radius of curvature
  • Evolute and involute
  • Envelope of a family of curves
  • Contact of plane curves
  • Geometric optimization problems
  • Physics optimization problems
  • Economic optimization problems
  • Van der Waals equation