Chapter 14 – Partial Derivatives 14.3 Partial Derivatives 1 Objectives:  Understand the various aspects of partial derivatives Dr. Erickson

Partial Derivative w.r.t. x at (a, b) In general, if f is a function of two variables x and y, suppose we let only x vary while keeping y fixed, say y = b, where b is a constant. Then, we are really considering a function of a single variable x g(x) = f(x, b) 14.3 Partial Derivatives2Dr. Erickson

Partial Derivative w.r.t. x at (a, b) If g has a derivative at a, we call it the partial derivative of f with respect to x at (a, b). We denote it by: f x (a, b) 14.3 Partial Derivatives3Dr. Erickson

Partial Derivative w.r.t. x at (a, b) So we have, By using the definition of derivative, this equation becomes 14.3 Partial Derivatives4Dr. Erickson

Partial Derivative w.r.t. y at (a, b) Similarly, the partial derivative of f with respect to y at (a, b), denoted by f y (a, b), is obtained by: ◦ Keeping x fixed (x = a) ◦ Finding the ordinary derivative at b of the function G(y) = f(a, y) 14.3 Partial Derivatives5Dr. Erickson

Partial Derivative w.r.t. y at (a, b) So we have, 14.3 Partial Derivatives6Dr. Erickson

Definition - Partial Derivatives If we now let the point (a, b) vary in Equations 2 and 3, f x and f y become functions of two variables Partial Derivatives7Dr. Erickson

Notation for Partial Derivatives If z = f (x,y), we can write 14.3 Partial Derivatives8Dr. Erickson

Rule for finding Partial Derivatives z = f (x,y) To find f x, regard y as a constant and differentiate f (x,y) w.r.t. x. To find f y, regard x as a constant and differentiate f (x,y) w.r.t. y Partial Derivatives9Dr. Erickson

Example 1 – pg. 912 # 16 Find the first partial derivatives of the function Partial Derivatives10Dr. Erickson

Example 2 Find the first partial derivatives of the function Partial Derivatives11Dr. Erickson

Function of more than Two Variables A function of three variables has the partial derivative w.r.t. x is defined as and is found by treating y and z as constants and differentiating the function w.r.t. x 14.3 Partial Derivatives12Dr. Erickson

Example 3 Find the first partial derivatives of the function Partial Derivatives13Dr. Erickson

Example 4 Find the first partial derivatives of the function Partial Derivatives14Dr. Erickson

Higher Derivatives If f is a function of two variables, then its partial derivatives f x and f y are also functions of two variables. So, we can consider their partial derivatives (f x ) x, (f x ) y, (f y ) x, (f y ) y These are called the second partial derivatives of f Partial Derivatives15Dr. Erickson

Notation 14.3 Partial Derivatives16Dr. Erickson

Example 5 Use implicit differentiation to find  z/  x and  z/  y Partial Derivatives17Dr. Erickson

Example 6 – pg. 913 # 54 Find all the second partial derivatives Partial Derivatives18Dr. Erickson

Example 7 Find the indicated partial derivative Partial Derivatives19Dr. Erickson

Clairaut’s Theorem 14.3 Partial Derivatives20Dr. Erickson

Example 8 – pg. 913 # 70 Find the indicated partial derivative Partial Derivatives21Dr. Erickson

More Examples The video examples below are from section 14.3 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 3 Example 3 ◦ Example 4 Example 4 ◦ Example 7 Example Partial Derivatives22Dr. Erickson

Demonstrations Feel free to explore these demonstrations below. Partial Derivatives in 3D Laplace's Equation on a Circle Laplace's Equation on a Square 14.3 Partial Derivatives23Dr. Erickson