chain - CHAIN RULE Math21a, O. Knill HOMEWORK Section 13.5:...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CHAIN RULE Math21a, O. Knill HOMEWORK Section 13.5: 2,6, 12, 44, 46 1D CHAIN RULE. If f and g are functions of one variable t , then d/dtf ( g ( t )) = f ( g ( t )) g ( t ). For example, d/dt sin(log( t )) = cos(log( t )) /t . FINDING DERIVATIVES. Also the 1D chain rule was useful. For example, to find the derivative of log( x ) we can write 1 = d/dx exp( log ( x ) = d/dx exp(log( x )) = log ( x ) x so that log ( x ) = 1 /x . An other example: to find arccos ( x ), we write 1 = d/dx cos(arccos( x )) = sin(arccos( x ))arccos ( x ) = radicalBig 1 sin 2 (arccos( x )) arccos ( x ) = 1 x 2 arccos ( x ) so that arccos ( x ) = 1 / 1 x 2 . GRADIENT. Define f ( x, y ) = ( f x ( x, y ) , f y ( x, y )). It is called the gradient of f . It is the natural derivative of a function of several variables and a vector. THE CHAIN RULE. If vector r ( t ) is curve in space and f is a function of three variables, we get a function of one variables t mapsto f ( vector r ( t )). The chain rule is d/dtf ( vector r ( t )) = f ( vector r ( t )) vector r ( t ) WRITING IT OUT. The chain rule is, when written out d dt f ( x ( t ) , y ( t )) = f x ( x ( t ) , y ( t )) x ( t ) + f y ( x ( t ) , y ( t )) y ( t ) . EXAMPLE. Let z = sin( x + 2 y ), where x and y are functions of t : x = e t , y = cos( t ). What is dz dt ? Here, z = f ( x, y ) = sin( x + 2 y ), z x = cos( x + 2 y ), and z y = 2 cos( x + 2 y ) and dx dt = e t , dy dt = sin( t ) and dz dt = cos( x + 2 y ) e t 2 cos( x + 2 y )sin( t )....
View Full Document

Ask a homework question - tutors are online