Partial Derivatives and Revision of Important Concepts

Introduction

• Welcome to Pradhikari Academy channel.
• Topic: Van Shot on Partial Derivatives for quick revision.
• Detailed lectures already available for in-depth understanding.
• Goal: Quick revision for solving related questions easily.

Basic Formulas

Derivatives (12th Grade Formulas)

• Power Rule: (\text{d/dx}(x^n) = nx^{n-1})
• Trigonometric Functions:
  • (\text{d/dx}(\sin x) = \cos x)
  • (\text{d/dx}(\cos x) = -\sin x)
  • (\text{d/dx}(\tan x) = \sec^2 x)
  • (\text{d/dx}(\cot x) = -\csc^2 x)
  • (\text{d/dx}(\sec x) = \sec x \tan x)
  • (\text{d/dx}(\csc x) = -\csc x \cot x)
• Logarithmic and Exponential Functions:
  • (\text{d/dx}(e^x) = e^x)
  • (\text{d/dx}(a^x) = a^x \ln(a))
  • (\text{d/dx}(\ln x) = \frac{1}{x})
• Special Functions:
  • (\text{d/dx}(x^2) = 2x)
  • (\text{d/dx}( ext{constant}) = 0)
  • (\text{d/dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}})

Product Rule

• For functions (u(x)) and (v(x)):
  • (\text{d/dx}(u \cdot v) = u'v + uv')

Quick Example Using Product Rule

• Given: (u = \sin x), (v = x^2)
  • (\text{d/dx}(u \cdot v) = \sin x \cdot 2x + x^2 \cdot \cos x)

Partial Derivatives

Basics of Partial Derivatives

• Understanding partial derivatives with respect to a variable while treating others as constants.
• Using the power rule for partial derivatives.

Example

• Given function: (u = x^3 + y^2)
• (\frac{\partial u}{\partial x} = 3x^2)
  • For variable (y), (\frac{\partial u}{\partial x}) ignores (y).
• (\frac{\partial u}{\partial y} = 2y)
  • For variable (x), (\frac{\partial u}{\partial y}) ignores (x).

More Examples and Explanation

• Apply partial differentiation rules systematically.
• Understand when terms become zero if treated as constants.
• Example: (\frac{\partial }{\partial x}(x^3 + y^2))
  • Result: (3x^2) since (y^2) treated as constant.

Advanced Topics

Euler’s Theorem for Homogeneous Functions

• Recognizing homogeneous functions and applying Euler's Theorem.
• Euler's Theorem: (x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y))
• Example: Prove if a given function is homogeneous.
  • Given: (f(x, y) = x^2 + y^2)
  • Proof: If homogeneous of degree 2, Euler’s Theorem is applied.
  • Steps to simplify and verify.

Composite Functions

• Discussing composite functions and their partial derivatives.
• Example format for solving such problems.
• Understanding the sequence of applying partial derivatives.

Variable to be Treated as Constant

• Discuss how to find partial derivatives when treating a variable as constant.
• Example: Function (u) in terms of (x) and (y) treating (z) as constant.

Practical Problem-Solving

• Step-by-step processes for dealing with real exam questions.
• Example problems demonstrating how to apply concepts.

Conclusion

• Quick revision covered partial derivatives and vital concepts.
• For detailed study, watch the full-length lectures.
• Encourage practice using the provided examples and methods.
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• Best wishes for exams and further studies.

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