In this lecture, we introduce the exponential function and study its main properties, derivatives, and applications in differentiation and integration.
We explain the relationship between the exponential and logarithmic functions, derive the formula for the derivative of e^x, and apply exponential functions in graph analysis and integration problems.
Topics covered:
• Definition of the exponential function e^x
• Relationship between e^x and ln(x)
• Basic properties of e^x
• Derivative of e^x
• Derivative of e^{g(x)} (chain rule)
• Applications of exponential functions
• Increasing/decreasing intervals
• Concavity and inflection points
• Exponential integrals
• Integrals involving substitution
• Integrals of the form f'(x)/f(x)
• Definite integrals with exponential functions
Several examples and exercises are discussed in detail.
Instructor: Prof. Shadi Shaqaqha
Department of Mathematics – Yarmouk University
MATH 101 – Calculus I
شرح الدالة الأسية e^x وخصائصها، ومشتقتها، وتطبيقاتها في التفاضل والتكامل، مع أمثلة متعددة على التكامل بالتعويض وتحليل المنحنيات.
00:00 Definition of the Exponential Function
04:44 Derivative of e^x
07:00 Graph Analysis Example
13:36 Exponential Integrals
