01_Ex 2.1

Exercise Questions

Question	Question Links
Q1. Find by definition, the derivatives w.r.t $x$ ,	2.1 Q-1
Q2. Find $\frac{d y}{d x}$ from first principals	2.1 Q-2

Overview

This exercise introduces the fundamental concept of differentiation by definition. Differentiation is a cornerstone of calculus, used extensively to study rates of change, slopes of curves, and optimization problems. The exercise emphasizes the manual derivation process, helping to strengthen the conceptual foundation of the derivative.

Key Concepts

Derivative by Definition:
The derivative of a function $f(x)$ is defined as:

f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}

Applications:
- Finding slopes of tangents to a curve.
- Solving real-world rate of change problems.
Common Functions:
- Polynomial functions (e.g., $x^n$ ).
- Root functions (e.g., $\sqrt{x}$ , $x^{1/3}$ ).
- Rational functions (e.g., $1/x$ , $1/x^m$ ).

Important Formulas

Below are the key formulas derived in this exercise:

Power Rule:
$\frac{d}{dx}(x^n) = nx^{n-1}$
Root Derivatives:
- $\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}$
- $\frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{-2/3}$
Reciprocal Functions:
- $\frac{d}{dx}(1/x) = -1/x^2$
- $\frac{d}{dx}(1/x^m) = -m/x^{m+1}$

Tips and Tricks

Simplify Before Differentiating:
For functions with complex fractions or roots, simplify the expression to make the limit evaluation easier.
Handle Rational Functions Carefully:
Rational functions often require finding a common denominator to evaluate the derivative.
Use the Power Rule:
Convert roots and reciprocals to powers for easier differentiation.

Summary

This exercise serves as a comprehensive introduction to finding derivatives by the first principles. It provides a step-by-step approach to handling different types of functions, reinforcing the rules and techniques essential for solving more advanced calculus problems.

Reference

By Great Science Academy:

By Sir Shahzad Sair: