second-order_partial_derivatives

second-order_partial_derivatives - Second-Order Partial...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
S econd- O rder P artial D erivatives Previously we have taken the partial derivative of a function f ( x , y ). But those partial derivatives were themselves functions and so we can take their partial derivatives. The Second-Order Partial Derivatives of z = f ( x , y ) 2 2 () x x zz x x f f xx x        2 yx y x f f xy x y  2 x y x y f f y x 2 2 yy y y f f y Example 1: Compute the second-order partial derivatives of f ( x , y ) = x 2 y + 5 x sin( y ). Solution: Notice that f x ( x , y ) = 2 xy + 5sin( y ) and f y ( x , y ) = x 2 + 5 x cos( y ). Thus, we have that f xx ( x , y ) = 2 y f xy ( x , y ) = 2 x + 5cos( y ) f yx ( x , y ) = 2 x + 5cos( y ) f yy ( x , y ) = –5 x sin( y ). Notice that in Example 1 above, we have that f xy ( x , y ) = f yx ( x , y ). Indeed, this is typically always the case. Thus, it does not matter if we take the partial derivative with respect to x first or with respect to y first. The Equality of Mixed Partial Derivatives If f xy ( x , y ) and f yx ( x , y ) are continuous at ( a , b ), an interior point of their domain, then (,) xy yx f ab f . 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
We can visualize this statement in the following diagram. z = f x,y z x y f x,y x,y xy x,y = yx x,y xx x,y yy x,y Figure 1: Equality of Mixed Partial Derivatives Example 2: Find all second-order partial derivatives of f ( x , y ) = ln(3 x + 5 y ). Solution: Notice that 3 (, ) 35 x fxy x y and 5 y x y . Thus, we have that  12 2 39 3 ( 3 5) 9 ( ( ) xx fx y x y x x       
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

second-order_partial_derivatives - Second-Order Partial...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online