notes-partials1 - OPMT 5701 Multivariable Calculus Partial...

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

View Full Document Right Arrow Icon
OPMT 5701 Multivariable Calculus Partial Derivatives Single variable calculus is really just a ”special case” of multivariable calculus. For the function y = f ( x ) , we assumed that y was the endogenous variable, x was the exogenous variable and everything else was a parameter. For example, given the equations y = a + bx or y = ax n we automatically treated a , b ,and n as constants and took the derivative of y with respect to x ( dy/dx ). However, what if we decided to treat x as a constant and take the derivative with respect to one of the other variables? Nothing precludes us from doing this. Consider the equation y = ax where dy dx = a Now suppose we f nd the derivative of y with respect to a , but TREAT x as the constant. Then dy da = x Here we just ”reversed” the roles played by a and x in our equation. Two Variable Case: let z = f ( x, y ) ,wh ichmeans z is a function of x and y .Inth i sc a s e z is the endoge- nous (dependent) variable and both x and y are the exogenous (independent) variables. To measure the the e f ect of a change in a single independent variable (x or y) on the dependent variable (z) we use what is known as the PARTIAL DERIVATIVE . The partial derivative of z with respect to x measures the instantaneous change in the function as x changes while HOLDING y constant . Similarly, we would hold x constant if we wanted to evaluate the e f ect of a change in y on z. Formally: z x is the ”partial derivative” of z with respect to x ,t r e a t in g y as a constant. Sometimes written as f x . z y is the ”partial derivative”
Background image of page 1

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

View Full DocumentRight Arrow Icon
The symbol (”bent over” lower case D) is called the ”partial” symbol. It is interpreted in exactly the same way as dy dx from single variable calculus. The symbol simply serves to remind us that there are other variables in the equation, but for the purposes of the current exercise, these other variables are held constant. EXAMPLES:
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 03/03/2012 for the course ECON 345 taught by Professor Wendywu during the Fall '11 term at Wilfred Laurier University .

Page1 / 8

notes-partials1 - OPMT 5701 Multivariable Calculus 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