{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

math handout

# math handout - ECONOMICS 100A MATHEMATICAL HANDOUT Fall...

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

ECONOMICS 100A MATHEMATICAL HANDOUT Fall 2007 Professor Mark Machina A. CALCULUS REVIEW Derivatives, Partial Derivatives and the Chain Rule * You should already know what a derivative is. We’ll use the expressions ƒ 6 ( x ) or d ƒ( x )/ dx for the derivative of the function ƒ( x ). To indicate the derivative of ƒ( x ) evaluated at the point x = x *, we’ll use the expressions ƒ 6 ( x *) or d ƒ( x *)/ dx . When we have a function of more than one variable, we can consider its derivatives with respect to each of the variables, that is, each of its partial derivatives . We use the expressions: ƒ( x 1 , x 2 )/ x 1 and ƒ 1 ( x 1 , x 2 ) interchangeably to denote the partial derivative of ƒ( x 1 , x 2 ) with respect to its first argument (that is, with respect to x 1 ). To calculate this, just hold x 2 fixed (treat it as a constant) so that ƒ( x 1 , x 2 ) may be thought of as a function of x 1 alone, and differentiate it with respect to x 1 . The notation for partial derivatives with respect to x 2 (or in the general case, with respect to x i ) is analogous. For example, if ƒ( x 1 , x 2 ) = x 1 2 x 2 + 3 x 1 , we have: ƒ( x 1 , x 2 )/ x 1 = 2 x 1 · x 2 + 3 and ƒ( x 1 , x 2 )/ x 2 = x 1 2 The normal vector of a function ƒ( x 1 ,..., x n ) at the point ( x 1 ,..., x n ) is just the vector (i.e., ordered list) of its n partial derivatives at that point, that is, the vector: ( ) 1 1 1 1 1 2 1 1 1 2 ƒ( ,..., ) ƒ( ,..., ) ƒ( ,..., ) , ,..., ƒ ( ,..., ),ƒ ( ,..., ),...,ƒ ( ,..., ) n n n n n n n n x x x x x x x x x x x x x x x = Normal vectors play a key role in the conditions for unconstrained and constrained optimization. The chain rule gives the derivative for a “function of a function.” Thus, if ƒ( x ) g ( h ( x )), then ƒ 6 ( x ) = g 6 ( h ( x )) h 6 ( x ) The chain rule also applies to taking partial derivatives. For example, if ƒ( x 1 , x 2 ) g ( h ( x 1 , x 2 )) then 1 2 1 2 1 2 1 1 ƒ( , ) ( , ) ( ( , )) x x h x x g h x x x x = Similarly, if ƒ( x 1 , x 2 ) g ( h ( x 1 , x 2 ), k ( x 1 , x 2 )) then: 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 2 1 1 1 ƒ( , ) ( , ) ( , ) ( ( , ), ( , )) ( ( , ), ( , )) x x h x x k x x g h x x k x x g h x x k x x x x x = + The second derivative of the function ƒ( x ) is written: ƒ Î ( x ) or d 2 ƒ( x )/ dx 2 and it is obtained by differentiating the function ƒ( x ) twice with respect to x (if you want the value of ƒ Î ( ) at some point x *, don’t substitute in x * until after you’ve differentiated twice). * If the material in this section is not already familiar to you, you may have trouble on both midterms and the final.

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

View Full Document