04_Ex 2.4

Exercise Questions

Questions	Question Links
Q1. Find $\frac{d y}{d x}$ by making suitable substitutions…	2.4 Q-1
Q2. Find $\frac{d y}{d x}$ if…	2.4 Q-2
Q3. Find $\frac{d y}{d x}$ of the following parametric functions.	2.4 Q-3
Q4. Prove that $y \frac{d y}{d x}+x=0$ if	2.4 Q-4
Q5. Differentiate…	2.4 Q-5

Overview

This exercise focuses on differentiation, emphasizing the use of substitution and chain rule to solve complex functions. Students will learn to handle multi-variable dependencies, simplify expressions, and differentiate nested functions efficiently. Mastery of these techniques is crucial for advanced calculus and its applications in physics, engineering, and mathematical modeling.

Key Concepts

Substitution Method: Rewriting functions using intermediate variables to simplify differentiation.
Chain Rule: Calculating derivatives of composite functions by breaking them into simpler parts.
Simplification: Utilizing algebraic manipulation to simplify derivative expressions.
Implicit Differentiation: Solving equations where $y$ is defined implicitly in terms of $x$ .

Important Formulas

Chain Rule:
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
Quotient Rule:
$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
Square Root Derivative:
$\frac{d}{dx} \sqrt{u} = \frac{1}{2\sqrt{u}} \cdot \frac{du}{dx}$

Tips and Tricks

Always simplify expressions before differentiating to reduce complexity.
When using substitution, clearly define and differentiate the intermediate variable.
Watch for common patterns like $\sqrt{x}$ and $x^{n}$ ; their derivatives often appear across multiple problems.
Break down multi-variable dependencies into smaller parts for better clarity and accuracy.
Be consistent with the chain rule when dealing with composite functions.

Summary

This exercise hones skills in advanced differentiation, focusing on composite functions, implicit differentiation, and the application of substitution. It reinforces understanding of core techniques such as the chain rule and quotient rule, which are essential for solving real-world problems involving rate of change and optimization.

Reference

By Great Science Academy:

By Sir Shahzad Sair: