11. Implicit Differentiation

What is 𝒊𝒎𝒑𝒍𝒊𝒄𝒊𝒕 differentiation? We'll see that if we restrict the domain and range of any function sufficiently, then (almost) any curve can be considered the graph of a function - even those that fail the so-called "vertical line test" you may have learned in a precalculus class. We'll use - surprisingly - implicit differentiation to prove the power rule in full generality: when the exponent can be any real number whatsoever.

This is the eleventh in a sequence of supplementary videos for a calculus class that I teach based on Chapters 1, 2, 3, and π of my book 𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠.

𝐹𝑢𝑙𝑙 𝐹𝑟𝑜𝑛𝑡𝑎𝑙 𝐶𝑎𝑙𝑐𝑢𝑙𝑢𝑠,
𝑇ℎ𝑒 𝐷𝑎𝑟𝑘 𝐴𝑟𝑡 𝑜𝑓 𝐿𝑖𝑛𝑒𝑎𝑟 𝐴𝑙𝑔𝑒𝑏𝑟𝑎,
𝑃𝑟𝑒𝑐𝑎𝑙𝑐𝑢𝑙𝑢𝑠 𝑀𝑎𝑑𝑒 𝐷𝑖𝑓𝑓𝑖𝑐𝑢𝑙𝑡,
𝐿𝑜𝑏𝑎𝑐ℎ𝑒𝑣𝑠𝑘𝑖 𝐼𝑙𝑙𝑢𝑚𝑖𝑛𝑎𝑡𝑒𝑑.

The first three are available for sale as paperbacks at Amazon, and as pdfs at Lulu. (The Lobachevski book is available at Amazon and the American Mathematical Society)

0:00 Non-functional relationships...
2:38 ...are functional when viewed through a small window.
5:27 Implicit Differentiation
16:10 dy/dx in terms of x AND y
19:35 A related example
25:07 The Power Rule for *all* real powers