02_Ex 2.2

This is a short exercise but for the sake of consistency in navigation, it has an Introduction page

Exercise Questions

Question	Question Links
Q1. Find from the first principles, the derivative…	2.2 Q-1

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Overview

This exercise involves calculating derivatives using the first principles for different algebraic expressions. These types of problems are fundamental in calculus and help build a deeper understanding of how derivatives are derived. The first principle of derivatives involves finding the limit of the difference quotient as the change in the independent variable approaches zero.

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Key Concepts

• Derivative: The derivative of a function represents the rate of change of the function with respect to its independent variable.

• First Principles: This approach calculates the derivative by taking the limit of the difference quotient: $f'(x) = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}$

• Binomial Expansion: In some cases, binomial expansion or series approximation is used to simplify expressions when calculating derivatives.

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Important Formulas

• The general formula for the derivative using first principles: $f'(x) = \lim_{\delta x \to 0} \frac{f(x + \delta x) - f(x)}{\delta x}$

• Binomial expansion for small changes: $(a + b)^n = a^n + n a^{n-1}b + \cdots$

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Tips and Tricks

• When applying first principles, carefully expand the expressions and simplify terms before taking the limit.
• For binomial expansions, ensure all terms are considered to the necessary degree, especially when dealing with powers or small increments.
• Focus on terms that remain as $\delta x \to 0$, as others will vanish.

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Summary

This exercise demonstrates the application of the first principle to find the derivative of various algebraic functions. The key steps involve expanding the expression for $f(x + \delta x)$, simplifying, and then taking the limit as $\delta x \to 0$. The examples provided cover both polynomial and rational functions, showing how different approaches, like binomial expansion, can be used to simplify the calculations.

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Reference

By Great Science Academy:

By Sir Shahzad Sair:

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