Notes on Derivative of Implicit Functions

Introduction

• Lecture by Neha, Math Mentor
• Topic: Derivative of Implicit Functions (Part 3 of Differentiation)
• Links for previous videos and playlists available in the description box.

Implicit Functions

• Definition: A function involving two variables (x and y) that cannot be expressed entirely in terms of one variable.
• Examples:
  • Example 1: Trying to express y in terms of x results in mixed variables.
  • Example 2: Cannot express x in terms of y entirely.

Differentiating Implicit Functions

1. Basic Principle: Differentiate with respect to x without trying to isolate y or x.
2. Use of Chain Rule: Important for differentiating terms involving y.

Example 1

• Given function: (F(x,y) = 0)
• Steps:
  • Differentiate term by term with respect to x.
  • Apply product rule for terms involving both x and y.
  • Collect terms containing (dy/dx) on one side.
  • Final result: (dy/dx = \frac{\sec^2 x - y}{x} )

Example 2

• Given function: (G(x,y) = 0)
• Steps:
  • Differentiate using product rule and chain rule.
  • Collect terms involving (dy/dx).
  • Final result: (dy/dx = \frac{\sin(xy)}{\sin(2y) - x \sin(xy)} )

Example 3

• Given function includes terms like (x^3 y^2) and logarithmic functions.
• Steps:
  • Use product rule for (x^3y^2).
  • Differentiate logarithmic and exponential functions accordingly.
  • Collect all (dy/dx) terms.
  • Final result: (dy/dx = \frac{Numerator}{Denominator})

Example 4

• Implicit differentiation of a more complex function.
• Steps:
  • Differentiate with respect to x, applying chain rule for square roots.
  • Collect terms involving (dy/dx).
  • Final result: (dy/dx = \frac{\sqrt{1 - y^2}}{1 - x^2})

Conclusion

• Importance of implicit differentiation in calculus.
• Encouragement to practice and understand the concept further.
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