Derivatives Using First Principles

This video tutorial explains how to derive the function e^x using the first principles of calculus. It begins with an introduction to derivatives and their importance, followed by a detailed explanation of the limit definition of a derivative. The instructor then applies this definition to e^x, demonstrating each step of the process, including transformations and simplifications. The video concludes with a proof that the derivative of e^x is indeed e^x, emphasizing the importance of continuous learning.

What is the main topic discussed in the video?

• Applications of calculus

• Derivatives using first principles

• Limits and continuity

• Integration techniques

Which function's derivative is being explored using the first principle?

• sin(x)

• cos(x)

• e^x

• ln(x)

What is the definition of a derivative according to the video?

• The limit as h approaches zero of (f(x+h) - f(x))/h

• The limit of a function as x approaches infinity

• The rate of change of a function

• The integral of a function over a given interval

What is the first step in applying the first principle to find the derivative of e^x?

• Simplify the expression

• Apply the laws of exponents

• Integrate the function

• Use the definition of a derivative

What transformation is introduced to simplify the limit expression?

• Substituting n for e^h - 1

• Using trigonometric identities

• Applying integration by parts

• Differentiating both sides

What does the transformation imply about the behavior of n as h approaches zero?

• n approaches infinity

• n remains constant

• n approaches zero

• n becomes undefined

What is the final result of the derivative of e^x using the first principle?

• e^x

• x^e

• ln(x)

• 1/x