This document demonstrates the calculation of the partial derivative of a two-dimensional Gaussian function with respect to x.
The step-by-step solution shows the application of differentiation rules in multivariable calculus.
Original Function
\[f(x, y) = e^{-(x^2 + y^2)}\]
This is a two-dimensional Gaussian function, which represents a bell-shaped surface centered at the origin (0,0).
Partial Derivative with Respect to x
\[\begin{align*}
\frac{\partial f}{\partial x} &= \frac{\partial}{\partial x}f(x,y)\\
&= \frac{\partial}{\partial x}\left(e^{-(x^2 + y^2)}\right)\\
&= e^{-(x^2 + y^2)} \frac{\partial}{\partial x}\left[-(x^2 + y^2)\right]\\
&= e^{-(x^2 + y^2)} \left[-\left(\frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial x}(y^2)\right)\right]\\
&= e^{-(x^2 + y^2)} \left[-\left(2x + 0\right)\right]\\
&= -2x e^{-(x^2 + y^2)}
\end{align*}\]