math10c - The Chain Rule So far we have considered partial...

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

View Full Document Right Arrow Icon
T he C hain R ule So far we have considered partial derivative and directional derivatives. The goal of this section is to address the following question: What is the derivative of a function f ( x , y ) where both x and y are functions of other variables? We begin with the simple case where x and y are both functions of just a single variable, t . That is, x = g ( t ) and y = h ( t ). This means that z = f ( x , y ) = f ( g ( t ), h ( t )). Thus, at the end of the day, z is actually just a function of t . It might help to see a tangible example. Suppose z = f ( x , y ) = x 2 + xy 3 , x = g ( t ) = t + 1 and y = h ( t ) = t 2 . If we substitute in for x and y , then we have that z = f ( x , y ) = f ( g ( t ), h ( t )) = ( t + 1) 2 + ( t + 1)( t 2 ) 3 = ( t + 1)( t 6 + t + 1). We can take the derivative of this function with respect to t just like we did when we first learned about derivatives, since this is now just a function of only one-variable, t . Thus, we see how we can take the derivative. However, this approach does seem a bit cumbersome. Let us try to develop a more concise way to express dz / dt in terms of derivatives of x ’s and y ’s. Using the notation from the section on local linearity, if x = x ( t ) and y = y ( t ), then we have that dx x t dt   and dy y t dt . And if z = f ( x , y ), then zz zxy x y    . Substituting in D x and D y , we have zdx zdy ztt x dt y dt x dt y dt  t   . Dividing both sides by D t , we have d xz d tx d ty d y t   . But as t ö 0, we have that dz z dx z dy dt x dt y dt  . We record this as the following: The Chain Rule for z = f ( x , y ), x = g ( t ), y = h ( t ) If z = f ( x , y ), x = g ( t ), and y = h ( t ) are differentiable functions, then dz z dx z dy dt x dt y dt 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
Example 1: Suppose f ( x , y ) = e x cos( y ) and x = t 3 and y = 3 t – 1. Compute dz / dt . Solution: Notice that cos( ) x z ey x , sin( ) x z y  , 2 3 dx t dt , and 3 dy dt . Putting this all together (remembering to substitute in for x and y with functions of t ), we have that
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.

This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

Page1 / 6

math10c - The Chain Rule So far we have considered 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