Partial Derivatives and the Gradient. Definition of Partial Derivative.

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Presentation transcript:

Partial Derivatives and the Gradient

Definition of Partial Derivative

Geometric Interpretation (x 0,y 0 ) x y z Slope = f x (x 0,y 0 )

Geometric Interpretation x y z Slope = f y (x 0,y 0 ) (x 0,y 0 )

Evaluating Partial Derivatives from the Definition

Example

Definition of Partial Derivative

Leibniz’s Notation

Example

Examples

Example r R h

Mixed Partials

Example

Remark

Definition of Gradient

Example

Does Partial Differentiability Imply Continuity?

Gradient Formulas