Differentiation: Definition, Formulas, Examples, and Facts

Introduction

• Differentiation in mathematics involves finding the derivative or rate of change of a function.
• Practical differentiation uses algebraic manipulations, basic derivatives, and rules of operation.

Basic Derivatives

1. Algebraic Functions: D(x^n) = n*x^(n-1), where n is any real number.
2. Trigonometric Functions:
   • D(sin x) = cos x
   • D(cos x) = -sin x
3. Exponential Functions: D(e^x) = e^x*

Rules of Differentiation

• Sum Rule: D(af + bg) = aDf + bDg
• Product Rule: D(fg) = fDg + gDf
• Quotient Rule: D(f/g) = (gDf - fDg)/g^2
• Chain Rule:
  • The derivative of a composite function f(g(x)) is Df(g(x)) * Dg(x).
  • Example: If f(x) = sin x^2, then D(sin x^2) = cos(x^2) * 2x.

Notable Mathematicians

• Gottfried Wilhelm Leibniz: Introduced d/dx notation for differentiation.
• Pierre de Fermat and Tullio Levi-Civita: Contributed to the development of differentiation.

Application in Calculus

• Differential Calculus: Concerned with the calculation of instantaneous rates of change.
• Integral Calculus: Involves summation of infinitely small factors to determine whole areas under curves.
• Fundamental Theorem of Calculus: Establishes that differentiation and integration are inverse operations.

History and Development

• Isaac Newton and Gottfried Wilhelm Leibniz are credited with the independent development of calculus in the 17th century.
• Calculus is foundational for physics, chemistry, economics, and other sciences.

Calculating Curves and Areas

• Roots of calculus date back to ancient geometry problems.
• Cavalieri and Descartes made significant contributions to the development of calculus.

Calculating Velocities and Slopes

• Calculus allows calculation of velocities (instantaneous rate of change) and slopes of curves.

Differentiation and Integration

• Differentiation involves finding the derivative of a function.
• Integration, the inverse of differentiation, involves finding areas under curves.
• Applications include graphing curves, determining maximum/minimum points, and physical interpretations in sciences.

Conclusion

• Differentiation, along with integration, forms the basis of calculus and has wide applications across various fields.

This summary provides a comprehensive overview of differentiation, its rules, notable contributors, and its importance in calculus.