List of basic formulas for finding derivatives of complex functions of several variables:

  1. Chain rule for functions of two variables:
    If ( z = f(x, y) ) and ( x = g(t) ), ( y = h(t) ), then the derivative of ( z ) with respect to ( t ) is calculated as:
    [
    \frac{dz}{dt} = \frac{\partial f}{\partial x} \cdot \frac{dx}{dt} + \frac{\partial f}{\partial y} \cdot \frac{dy}{dt}
    ]
  2. Partial derivatives:
    For a function (z = f(x, y)):
  • Partial derivative with respect to ( x ):
    [
    \frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) – f(x, y)}{\Delta x}
    ]
  • Partial derivative with respect to ( y ):
    [
    \frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, y + \Delta y) – f(x, y)}{\Delta y}
    ]
  1. Total derivative:
    Total derivative of a function ( z = f(x, y) ) with respect to a variable ( t ):
    [
    dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy
    ]
  2. Second derivatives:
  • Second partial derivative with respect to ( x ):
    [
    \frac{\partial^2 f}{\partial x^2}
    ]
  • Second partial derivative with respect to ( y ):
    [
    \frac{\partial^2 f}{\partial y^2}
    ]
  • Mixed derivative:
    [
    \frac{\partial^2 f}{\partial x \partial y}
    ]

These formulas will help you in working with derivatives of complex functions of several variables.

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