Lecture on Chain Rule in Calculus

Introduction to Chain Rule

• Core principle in calculus for taking derivatives of complex functions.
• Initially daunting, but becomes simpler and intuitive with practice.

Example Function

• Consider function ( h(x) = \sin(x^2) ).
• The task is to find ( h'(x) ) or the derivative of ( h ) with respect to ( x ).
• Notation: ( h'(x) = \frac{dh}{dx} ).

Derivative via Chain Rule

• Chain rule applies when function is a composition of more than one function.
• Steps to apply Chain Rule:
  1. Differentiate outer function.
  2. Multiply by the derivative of the inner function.

Thought Experiment

• Derivative of ( x^2 ) with respect to ( x ) is ( 2x ).
• Derivative of ( a^2 ) with respect to ( a ) is ( 2a ).
• Derivative with respect to ( \sin(x) ) of ( \sin(x^2) ):
  • Replace variable with ( \sin(x) ).
  • Result: ( 2\sin(x) ).

Applying Chain Rule to ( h(x) = \sin(x^2) )

• Derivative of outer function ( x^2 ) with respect to inner ( \sin(x) ):
  • ( 2\sin(x) ).
• Multiply by derivative of inner function (( \sin(x) ) with respect to ( x )):
  • Derivative of ( \sin(x) ) is ( \cos(x) ).
• Combine results:
  • ( h'(x) = 2\sin(x) \cdot \cos(x) ).

Intuition and Notation

• Treat differentials ( dx ), ( d\sin(x) ) like fractions for intuition.
• Not rigorous but helps in understanding and visualizing the process.

Conclusion

• Understanding requires practicing more examples.
• Further examples and abstraction in future videos.