Imagine a surface [math]z=f\left(x,y\right)[/math] and consider a particular point in its domain, [math]\left(x_0,y_0\right)[/math]. As one "moves through" [math]\left(x_0,y_0\right)[/math] in the [math]xy[/math]-plane, traveling in the positive [math]x[/math]-direction, the value of [math]f\left(x,y\right)[/math] changes - the rate at which [math]f[/math] changes is called [i]the partial derivative of [/i][math]f[/math][i] with respect to [/i][math]x[/math][i] at[/i] [math]\left(x_0,y_0\right)[/math], and is denoted [math]f_x\left(x_0,y_0\right)[/math]. Similarly, one finds the partial derivative of [math]f[/math] with respect to [math]y[/math], [math]f_y\left(x_0,y_0\right)[/math] by moving through [math]\left(x_0,y_0\right)[/math] in the positive [math]y[/math] direction.[br][br]In the interactive figure, select (or create) a surface [math]z=f\left(x,y\right)[/math], and use the sliders to position the point [math]\left(x_0,y_0\right)[/math]. By viewing cross-sections of the surface, you can see the partial derivatives of [math]f[/math] at that point as slopes of certain lines tangent to the surface through the point [math]\left(x_0,y_0,f\left(x_{0,}y_0\right)\right)[/math].

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]