91Ó°ÊÓ

Partial derivatives are fundamental in understanding how a surface changes in different directions. When dealing with a function of two variables, like \( z = \tan^{-1}(xy) \), you take partial derivatives to see how changes in \(x\) or \(y\) affect \(z\) independently.
To find the partial derivatives, we hold one variable constant while differentiating with respect to the other. Here, the partial derivative with respect to \(x\), denoted \( \frac{\partial z}{\partial x} \), assumes \(y\) is constant. Similarly, \( \frac{\partial z}{\partial y} \) considers \(x\) constant. \(
\)For our surface, \( \frac{\partial z}{\partial x} = \frac{y}{1+(xy)^2} \) and \( \frac{\partial z}{\partial y} = \frac{x}{1+(xy)^2} \). These provide the rates of change in the surface’s height in the \(x\) and \(y\) directions, respectively.

The gradient of a surface at a point is a vector that combines the partial derivatives. It points in the direction of the steepest ascent on the surface and its magnitude represents how steep this ascent is.
For the function \( z = \tan^{-1}(xy) \), the gradient at a point \((x, y)\) is expressed as the vector \( abla z = \left( \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y} \right) \).
At the point \((1,1)\), both partial derivatives evaluated to \( \frac{1}{2} \). This gives the gradient vector \( \left( \frac{1}{2}, \frac{1}{2} \right) \) at the point \((1,1, \pi/4)\). This vector tells us that the surface rises equally in the \(x\) and \(y\) directions at this point.

The surface equation describes a surface in three-dimensional space. In this context, the surface is given by \( z = \tan^{-1}(xy) \).
We are interested in finding the equation of the tangent plane to this surface at a specific point \((1,1, \pi/4)\). A tangent plane gives us the linear approximation of the surface near the point. It touches the surface at exactly one point without cutting through it.
To find this plane, we use the partial derivatives at the point, as these are crucial for constructing the equation. The general tangent plane formula is:

• \( \frac{\partial z}{\partial x} (x - x_0) + \frac{\partial z}{\partial y} (y - y_0) = z - z_0 \)

For our surface, substituting the values gives us:

• \( \frac{1}{2}(x-1) + \frac{1}{2}(y-1) = z - \frac{\pi}{4} \)

This simplifies to \( \frac{1}{2}x + \frac{1}{2}y - z = -\frac{\pi}{4} + \frac{3}{2} \), which is the precise equation of our tangent plane.