Dated: 30-10-2024
Geometric Meaning of Partial Derivative
Geometric Meaning of partial derivative
Imagine we have
\[z = f(x, y)\]
then the partial derivative with respect to \(x\) is denoted by
\[\frac{\partial z}{\partial x}\]
\[f_x\]
\[\frac{\partial f}{\partial x}\]
and with respect to \(y\), it is
\[\frac{\partial z}{\partial y}\]
\[f_y\]
\[\frac{\partial f}{\partial y}\]
Partial Derivatives
Let \(z = f(x, y)\) be a function of \(x\) and \(y\), then keeping \(y\) as a constant, change in \(f(x, y)\) due to change if \(x\) can be given by
\[\Delta z = f(x + \Delta x, y) - f(x, y)\]
If the following ratio
\[\frac{\Delta z}{\Delta x} = \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}\]
approaches to a limit \(\Delta x \to 0\) then
\[f_x(x, y) = \frac{\partial z}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x, y) - f(x, y)}{\Delta x}\]
This is called partial derivative with respect to \(x\).
Similarly, the partial derivative with respect to \(y\) will be
\[f_y(x, y) = \frac{\partial z}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x, \Delta y + y) - f(x, y)}{\Delta y}\]
Geometric Meaning of partial derivative
Imagine we have \(z = f(x, y)\) being a function of 2 variables and its graph is a surface.
Then the coordinates of any arbitrary point \(P\) at \((x_0, y_0)\) are \(P(x_0, y_0, f(x_0, z_0))\).
And if this point starts moving on the surface such that \(y\) remains constant then
On this curve, \(\frac{\partial z}{\partial x}\) is the partial derivative with respect to \(x\) since \(y\) is constant. This gives us the tangent slope to this curve.
Similarly, for \(y\), we have
Partial Derivatives of Higher Orders
\[\frac{\partial}{\partial x} \left(\frac{\partial f}{\partial x}\right) = \frac{\partial ^2 f}{\partial x^2} = \frac{\partial}{\partial x} f_x = f_{xx} = f_{x^2}\]
\[\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right) = \frac{\partial ^2 f}{\partial y \partial x} = \frac{\partial}{\partial y} f_x = f_{xy}\]
\[\frac{\partial}{\partial x} \left(\frac{\partial f}{\partial y}\right) = \frac{\partial ^2 f}{\partial x \partial y} = \frac{\partial}{\partial x} f_y = f_{yx}\]
\[\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial y}\right) = \frac{\partial ^2 f}{\partial y^2} = \frac{\partial}{\partial y} f_y = f_{yy} = f_{y^2}\]
The \(f_{xy}\) and \(f_{yx}\) are called mixed second partials and usually are not equal.
Example
\[ f(x, y) = x \cos y + y e^x \]
\[ \frac{\partial f}{\partial x} = \cos y + y e^x \]
\[ \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = -\sin y + e^x \]
\[ \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = y e^x \]
\[ f(x, y) = x \cos y + y e^x \]
\[ \frac{\partial f}{\partial y} = -x \sin y + e^x \]
\[ \frac{\partial^2 f}{\partial x \partial y} = -\sin y + e^x \]
\[ \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) = -x \cos y \]
Laplace's Equation
Let \(w=f(x, y, z)\) be a function then
\[\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0\]
It also works for functions of 2 variables.
Euler's Theorem or Mixed Derivative Theorem
If \(f(x, y)\) and its partial derivatives \(f_x, f_y, f_{xy}\) and \(f_{yx}\) are defined throughout an open region containing a point \((a, b)\) and are all continuous at \((a, b)\) then
\[f_{xy}(a, b) = f_{yx}(a, b)\]
Advantages
\[w = xy + \frac{e^y}{y^2 + 1}\]
The following suggests us to take partial derivative with respect to \(y\) first and then with respect to \(x\).
\[\frac{\partial ^ 2 w}{\partial x \partial y}\]
Using the theorem, if we partially derivate with respect to \(x\) first then finding the answer becomes more quicker.
\[\frac{\partial w}{\partial x} = y\]
\[\frac{\partial^2 w}{\partial y \partial x} = 1\]
References
Read more about notations and symbols.