Why partial derivatives aren't enough to define smooth change
In short: In single-variable calculus, a derivative tells you everything about a function's local behavior. For multivariable functions, the math is more complex. This video explores why simply having partial derivatives doesn't guarantee a function is truly differentiable.
The video examines the nuances of differentiability for functions of multiple variables. While partial derivatives measure change along specific axes, they fail to capture the full behavior of a surface. To be truly differentiable, a function must be locally linear—meaning it can be well-approximated by a tangent plane rather than just a set of independent slopes.
Dr. Trefor Bazett explains that differentiability is a stronger condition than the mere existence of partial derivatives. It requires that the linear approximation becomes increasingly accurate as you move closer to a specific point, ensuring the function behaves predictably from every possible direction of approach.
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- Summary by Intellect · source: What is differentiability for multivariable functions??
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