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The Multi-Variable Chain Rule: Derivatives of Compositions

Suppose that f(x,y) depends on two variables but that the x(t) and y(t) are themselves both functions of t. Then f(x(t), y(t)) is a composition of functions and the derivative of f with respect to t is computed via the multi-variable Chain Rule. In fact, this scenario is one of many different generalizations of the single variable chain rule. We can use arrow (or dependency) diagrams to illustrate the relationships between a bunch of multivariable functions and for each situation write down a chain rule that gives the slope. The idea in the example given here is that a small change in t results in a small change in both x and y which in turn result in a small change in f. **************************************************** YOUR TURN! Learning math requires more than just watching videos, so make sure you reflect, ask questions, and do lots of practice problems! **************************************************** ►Full Multivariable Calculus Playlist:    • Calculus III: Multivariable Calculus ...   **************************************************** Other Course Playlists: ►CALCULUS I:    • Calculus I (Limits, Derivative, Integ...   ► CALCULUS II:    • Calculus II (Integration Methods, Ser...   ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Log...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   *************************************************** ► Want to learn math effectively? Check out my "Learning Math" Series:    • 5 Tips To Make Math Practice Problems...   ►Want some cool math? Check out my "Cool Math" Series:    • Cool Math Series   ***************************************************** ►Follow me on Twitter:   / treforbazett   ***************************************************** This video was created by Dr. Trefor Bazett BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS & MERCH I LOVE: ► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett