MAT 1236 Calculus III Section 14.3 Partial Derivatives

HW.. WebAssign 14.3 Quiz 14.2, 14.3 Exam 1….

Preview Discuss the definitions of derivatives of functions with 2 variables Discuss how the concept is the extension of the 1 variable case

1 variable Vs 2 variables

Quotient Difference

Derivatives

Partial Derivatives

y is kept constant Partial Derivatives x is kept constant

For, we hold y constant For, we hold x constant How to find partial derivatives?

Example 1 Find

Example 1(a), (b)

Example 1(c), (d)

Example 2 Find

Example 2 (a)

Example 2 (b)

Example 3 (Implicit Diff.) Find z as a function in x and y

Higher Derivatives

1 variable Vs 2 variables: Higher Derivatives

Second Order Derivatives

Clairaut’s Theorem

Example 4 Find all the second order derivatives

Geometric Meanings

Example 5 Show that is a solution of Laplace’s Equation

Expectations Always start with one side and show that it equals to the other side. Normally, it is easier to start with the side with more complicated terms.

Quiz Take a look at the solutions Think about the grader’s comments Do not (get emotional and blame the grader) Talk to me in my office when you are NOT emotional