Section 14'1 The Partial Derivative

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Section 14.1 The Partial Derivative
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• The following contour plot shows the monthly
  payment as a function of the interest rate and
  loan amount

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• For a fixed interest rate the spacing between
  contours is even, why?
• Holding rate constant, is the rate of change
  positive or negative?
• Does the constant spacing hold for a fixed loan
  amount?

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Old stuff

• Let y f(x), then
• This gives us
• The slope of the tangent line at x a
• The instantaneous rate of change at x a

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New stuff

• Let z f(x, y)
• We can find partial derivates with respect to
  each of the input variables (we hold the other
  constant)
• The partial derivates at the point (a, b) is

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Interpretations

• Numerical
• It gives us the instantaneous rate of change of
  f(x,y) at (a,b) per unit increase in x with y
  held fixed at b (or in y with x held fixed at a)
• Graphical
• It gives us the slope of the tangent line at
  (a,b) in the x direction (or y direction)

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Estimating partial derivatives

• If

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• Suppose z f(x,y) is given by the table
• Estimate the following

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Notations for the partial derivatives

• If z f(x,y) we can write