91Ó°ÊÓ

In calculus, partial derivatives represent the rate at which a function changes as one of its variables changes while all other variables are held constant. For a function of two variables, like \( f(x, y) \), you can take the partial derivative with respect to \( x \) and \( y \).
To do this, you treat every other variable as if it were a constant.With our function \( f(x, y) = (x+y)/(x-y) \), we first have to calculate the partial derivatives with respect to each variable. Specifically, \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). These represent how \( f \) changes when either \( x \) or \( y \) changes.

• \( \frac{\partial f}{\partial x} = \frac{-2y}{(x-y)^2} \)
• \( \frac{\partial f}{\partial y} = \frac{-2x}{(x-y)^2} \)

By understanding these rates of change, we can better approximate function values near the point of interest.

The quotient rule is essential when differentiating a function that is the division of two other functions. When you have a function that looks like \( g(x) / h(x) \), the rule helps you find its derivative.Here's the formula for the quotient rule:\[\left( \frac{g}{h} \right)' = \frac{g' \cdot h - g \cdot h'}{h^2}\]In this problem, our function \( f(x, y) = \frac{x+y}{x-y} \) requires the quotient rule.
By substituting \( g = x + y \) and \( h = x - y \) into the formula, we differentiate with respect to \( x \) or \( y \) as needed.

• When finding \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant during differentiation.
• When finding \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant.

The quotient rule can be complex but following step by step helps manage this complexity.

The tangent plane is a linear approximation of a function at a given point. It provides a simple way to estimate function values close to this point with just a flat plane.For a function \( f(x, y) \), the equation for the tangent plane at a particular point \((a, b)\) is:\[L(x, y) = f(a, b) + \frac{\partial f}{\partial x}(a, b)(x-a) + \frac{\partial f}{\partial y}(a, b)(y-b)\]This formula uses the function value as well as its partial derivatives at the chosen point.

• In our example, with the point (3, 2), and \( f(3, 2) = 5 \), the tangent plane simplification helps us estimate nearby values.
• Thus, at (2.95, 2.05), the estimated function value using the tangent plane is very close to the actual value, approximately \( 4.9 \).

The tangent plane is especially useful for functions that are too complex to calculate exactly at every point, making approximation easier.