91Ó°ÊÓ

When dealing with multivariable functions, we often use partial derivatives to understand how the function changes with respect to one variable while keeping others constant. In this exercise, we have an equation involving both \(x\) and \(y\), so we are interested in its behavior with each variable separately.

• **Partial derivative with respect to \(x\):** This considers \(y\) as a constant and finds the rate of change as \(x\) varies. For the equation \(x^2 + xy + y^2 - 7 = 0\), the partial derivative with respect to \(x\) is \(2x + y + x\frac{dy}{dx}\).
• **Partial derivative with respect to \(y\):** This considers \(x\) as a constant and observes changes with varying \(y\). For the same equation, the partial derivative with respect to \(y\) is \(x + 2y\frac{dy}{dx}\).

Partial derivatives are crucial for implicit differentiation, which we use to find \(\frac{dy}{dx}\) in equations where \(y\) is not explicitly solved in terms of \(x\). Understanding how each variable influences the function is key.

The implicit function theorem is a fundamental concept in calculus, allowing us to differentiate functions even when they are not given explicitly. In our scenario, the equation \(x^2 + xy + y^2 - 7 = 0\) relates \(x\) and \(y\) implicitly.
This theorem states that if a function \(F(x, y) = 0\) is continuously differentiable near a point \((x_0, y_0)\), and the partial derivative of \(F\) with respect to \(y\) is non-zero at this point, then \(y\) can locally be a differentiable function of \(x\).
In simpler terms, at the point \((1, 2)\), because the partial derivative \(\frac{\partial F}{\partial y} = 2y\) is non-zero, the theorem assures us that \(y\) varies smoothly with \(x\). We combine both partial derivatives derived earlier to solve for \(\frac{dy}{dx}\), following the logic of the implicit function theorem. This is why we equate and manipulate them to eventually solve for the derivative.

Equations that describe curves on a plane, especially those involving multiple variables, sometimes define relationships implicitly. The given equation \(x^2 + xy + y^2 - 7 = 0\) represents such a curve. Unlike functions that express \(y\) explicitly in terms of \(x\), implicit equations describe relationships without isolating a variable.

• These equations describe level curves or contour lines, representing specific values on a surface or in space.
• Analyzing these geometric shapes or paths helps understand how variables depend on one another.

In the example provided, by examining the partial derivatives and using implicit differentiation, we find the slope of the curve at a given point. This slope \(\frac{dy}{dx}\) tells us the rate of change or steepness at the point \((1, 2)\) on the curve described by the equation. Understanding the equation of a curve in this way combines the beauty of geometry with the precision of calculus.