Differentiation - Basic Concepts

Introduction

• शुरुआत डेरिवेटिव्स और डिफरेंटशिएशन की
• सेशन में पेसिफिक डिफरेंटशिएशन
• Part 1: Basics of Differentiation
• Class 12th syllabus में शामिल

What is a Derivative?

• Derivative का मीनिंग: स्लोप (Slope of a Tangent to the Curve)
• Derivative = Slope of Tangent to Function’s Graph
• Mathematical language: अगर पॉइंट के कोऑर्डिनेट एक्स और एक्स प्लस एच हों तो:
  • Slope = (f(x+h) - f(x))/h जब h -> 0
  • Derivative notation: f'(x) या dy/dx
  • First Principle Method to find derivatives: lim(h -> 0) [(f(x+h) - f(x))/h]

Important Formulas

• Derivatives of basic functions using First Principle:
  • Slope of f(x) at point x is derived as: lim(h -> 0) [(f(x+h) - f(x))/h]

Examples Using First Principle Method

• Derivative of sin(x): Using first principle, derivative of sin(x) with respect to x is cos(x)
• Derivative of x^2: Using first principle, derivative is 2x

Basic Differentiation Formulas to Remember

Trigonometric Functions

• d/dx[sin x] = cos x
• d/dx[cos x] = -sin x
• d/dx[tan x] = sec² x
• d/dx[cot x] = -csc² x
• d/dx[sec x] = sec x tan x
• d/dx[csc x] = -csc x cot x

Inverse Trigonometric Functions

• d/dx[sin⁻¹ x] = 1/√(1 - x²)
• d/dx[cos⁻¹ x] = -1/√(1 - x²)
• d/dx[tan⁻¹ x] = 1/(1 + x²)
• d/dx[cot⁻¹ x] = -1/(1 + x²)
• d/dx[sec⁻¹ x] = 1/(|x|√(x² - 1))
• d/dx[csc⁻¹ x] = -1/(|x|√(x² - 1))

Exponential and Logarithmic Functions

• d/dx[a^x] = a^x ln(a)
• d/dx[e^x] = e^x
• d/dx[ln x] = 1/x
• d/dx[logₐ(x)] = 1/(x ln(a))

Polynomial Functions

• d/dx[x^n] = nx^(n-1)
• d/dx[k] = 0
• d/dx[√x] = 1/(2√x)

Theorems for Differentiation

Sum/Difference Rule

• d/dx[f(x) ± g(x)] = d/dx[f(x)] ± d/dx[g(x)]

Product Rule

• d/dx[f(x)g(x)] = f(x)d/dx[g(x)] + g(x)d/dx[f(x)]

Quotient Rule

• d/dx[f(x)/g(x)] = (g(x)d/dx[f(x)] - f(x)d/dx[g(x)]) / (g(x))²

Chain Rule

• If y = f(g(x)), then dy/dx = f'(g(x))*g'(x)

Advanced Differentiation Techniques

• Understanding and applying the above theorems
• Eg: If f(x) = e^(sin(x)), using chain rule:
  • f'(x) = e^(sin(x)) * cos(x)
• Practical application in simplifying complex differentiation problems

Practice and Application

• Various examples to practice using the mentioned rules
• Emphasis on attempting problems for mastery
• Encouragement to share solutions and actively participate in discussions

Conclusion

• Recapping the importance of understanding basics for mastering differentiation
• Future sessions to cover more advanced topics and applications in differentiation