This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 15.3. Partial Derivatives 15.4. Tangent Planes and Linear Approximation Partial derivative of z = f ( x,y ) with respect to x at ( a,b ) is denoted by f x ( a,b ) and it is f x ( a,b ) = ∂f ∂x ( a,b ) = lim h → f ( a + h,b )- f ( a,b ) h . Partial derivative of z = f ( x,y ) with respect to y at ( a,b ) is denoted by f y ( a,b ) and it is f y ( a,b ) = ∂f ∂y ( a,b ) = lim h → f ( a,b + h )- f ( a,b ) h . Rule for finding partial derivatives. (1) To find f x , consider y as a constant and differentiate z = f ( x,y ) wit respect to x . (2) To find f y , consider x as a constant and differentiate z = f ( x,y ) wit respect to y . By analogy we consider higher order partial derivatives: f xx ,f xy ,f yx ,f yy ,f xxy , etc. By analogy we consider partial derivatives of function w = f ( x,y,z ) of three and more variables w = f ( x 1 ,x 2 ,...,x n ). Examples. Find the first partial derivatives of the function: f ( x,y ) = y 5- 3 xy, f ( x,t ) = √ x ln( t ) , w = e v u + v 2 f ( x,y,z,t ) = xy 2 t + 2 z , f ( x,y ) = ln( x + q x 2 + y 2 ) , f ( x,y,z ) = xyz x + y + z Clairaut’s Theorem. Suppose z = f ( x,y ) is defined on a disk D containing ( a,b ). If both mixed second partial derivatives f xy ( x,y ) and f yx ( x,y ) are continuous on D , then f xy ( a,b ) = f yx ( a,b ) . Example. Verify the conclusions of Clairaut’s Theorem: u = ln q x 2 + y 2 , v = xe xy . Example. Find all second order partial derivatives: w = p u 2 + v 2 . 1 Geometric Interpretation of the First Order Partial Derivatives as Slopes of Tangent Lines. Linear Equation of a Tangent Plane. Given the equation z = f ( x,y ) of a surface S and a point ( a,b ) on the xy- plane in the domain of z = f ( x,y ). Then, P ( a,b,f ( a,b )) is a point on S ....
View Full Document