1 / 10

Section 15.3

Section 15.3. Partial Derivatives. PARTIAL DERIVATIVES. If f is a function of two variables, its partial derivatives are the functions f x and f y defined by. NOTATIONS FOR PARTIAL DERIVATIVES. If z = f ( x , y ), we write.

taylor-dillard
Télécharger la présentation

Section 15.3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 15.3 Partial Derivatives

  2. PARTIAL DERIVATIVES If f is a function of two variables, its partial derivatives are the functions fx and fy defined by

  3. NOTATIONS FOR PARTIAL DERIVATIVES If z = f (x, y), we write

  4. RULE FOR FINDING PARTIAL DERIVATIVES OF z = f (x, y) • To find fx , regard y as a constant and differentiate f (x, y) with respect to x. • To find fy , regard x as a constant and differentiate f (x, y) with respect to y.

  5. FUNCTIONS OF MORE THAN TWO VARIABLES There are analogous definitions for the partial derivatives of functions of three or more variables.

  6. GEOMETRIC INTERPRETATION OF PARTIAL DERIVATIVES Consider the surface S whose equation is z=f(x,y). The plane y = b intersects this surface in a plane curve C1. The value of fx(a, b) is the slope of the tangent line T1 to the curve at the point P(a, b, f(a, b)). Similarly, the plane x = a intersects the surface in a plane curve C2 and fy(a,b) is the slope of the tangent line T2 to the curve at the point P(a, b, f(a, b)).

  7. SECOND PARTIAL DERIVATIVES If f is a function of two variables, then its partial derivatives fx and fy are also functions of two variables, so we can consider their partial derivatives ( fx)x, ( fx)y, ( fy)x, and ( fy)y, which are called the second partial derivatives of f.

  8. NOTATION FOR THE SECOND PARTIAL DERIVATIVES If z = f (x, y), we use the following notation:

  9. CLAIRAUT’S THEOREM Suppose that f is defined on a disk D that contains the point (a, b). If the functions fxy and fyx are both continuous on D, then fxy(a, b) = fyx(a, b)

  10. HIGHER ORDER DERIVATIVES If f is a two variable function, partial derivatives of order 3 and higher can be defined. Some examples would be fxxx , fxyx , fxyyx , etc. Using Clairnaut’s Theorem, we can show that fxyy = fyxy = fyyx if these functions are continuous and fxxy = fxyx = fyxx if these functions are continuous.

More Related