List of basic formulas for finding derivatives of complex functions of several variables:
- 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}
] - 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}
]
- 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
] - 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.