partial_derivatives

# partial_derivatives - Partial Derivatives Recall in...

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P artial D erivatives Recall in one-variable calculus, we introduced the derivative of a function. The goal of this section (and the remainder of the course) is to take a derivative of a function with two (or more) variables. When there was only one variable, the derivative at a particular point had a clear interpretation: it was the instantaneous rate of change of the function at that point. Equivalently, it was the slope of the tangent line to the function at that point. But things are more complicated when we deal with two variables. There is no longer a single tangent line at a particular point on a surface. In order to deal with this potential problem, we shall define the partial derivative of a function at a point. Essentially, we take the derivative with respect to one of the variables (either x or y ), holding the other variable fixed. (That is, we think of it as a constant for the time being.) This has the effect of reducing our function of two variables to a function of only one variable and we can use our previous definition of the derivative to make sense of the situation. Partial Derivatives of f ( x , y ) With Respect to x and y For all points at which the limits exists, we define the partial derivatives of the function f ( x , y ) at the point ( a , b ) to be: 0 Rate of change of ( , ) with (, ) ( , (,) l im respect to at the point ( , ) x h fxy ) f ah b fa b fab xa b h   0 Rate of change of ( , ) with ) l respect to at the point ( , ) y h f ab h f ab ya b h Letting a and b vary, then we can define the partial derivative functions f x ( x , y ) and f y ( x , y ). Just like derivatives with only one variable, we have some short-hand notation that is used often. 1

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Short-hand Notation for Partial Derivatives If z = f ( x , y ), we can write the partial derivative functions as (, ) x z fxy x and y z y We can define the partial derivatives at a point ( a , b ) as (,) x ab z fab x and y z y
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## This note was uploaded on 09/07/2011 for the course MATH 10C taught by Professor Hohnhold during the Spring '07 term at UCSD.

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partial_derivatives - Partial Derivatives Recall in...

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