91Ó°ÊÓ

Partial derivatives are a fundamental concept when dealing with functions of multiple variables. They help us understand how a function changes as we vary just one of the variables, while keeping the others constant. Think of it like looking at a hill and trying to figure out how steep it is as you walk in different directions. That's what partial derivatives do for multidimensional surfaces. For a function of two variables, say \(F(x, y)\), the partial derivative with respect to \(x\) is denoted as \(\frac{\partial F}{\partial x}\), and it measures the rate at which \(F\) changes as we change only \(x\). Similarly, \(\frac{\partial F}{\partial y}\) measures the rate of change as we vary \(y\). In our exercise, the equation \(6x^{2} + 3xy + 2y^{2} + 17y - 6 = 0\) was used to find the partial derivatives, leading to:

• \(\frac{\partial F}{\partial x} = 12x + 3y\)
• \(\frac{\partial F}{\partial y} = 3x + 4y + 17\)

These derivatives were then evaluated at the point \((-1, 0)\) to assist in determining the slope of the tangent line.

The point-slope form of a line is a very handy formula for finding the equation of a line when you know the slope and a point through which the line passes. It's formulated as \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is any point on the line, and \(m\) is the slope. This form is particularly useful in this exercise because once the slope of the tangent line was calculated as \(\frac{6}{7}\) using the partial derivatives at \((-1, 0)\), applying point-slope form became straightforward:

• The tangent line equation is: \(y = \frac{6}{7}x + \frac{6}{7}\)
• For the normal line, whose slope is the negative reciprocal, \(-\frac{7}{6}\), the equation becomes: \(y = -\frac{7}{6}x - \frac{7}{6}\)

Using the point-slope form simplifies the process of determining these equations and is especially useful when dealing with curves.

Implicit differentiation allows us to find derivatives of relationships that are not easily solved for one variable in terms of another. Consider a curve given by an implicit equation like the one in our exercise, \(6x^{2}+3xy+2y^{2}+17y-6=0\). Solving for \(y\) in terms of \(x\) or vice versa can be quite complex, if not impossible. Instead, we use implicit differentiation, which involves taking the derivative of both sides of the equation with respect to a chosen variable, typically \(x\). This means applying derivatives directly to the function, treating \(y\) as a function of \(x\), which often gives us expressions involving \(\frac{dy}{dx}\). However, for this exercise, it’s the mixed nature and relationships of the variables that require partial derivatives for each variable:

• Find \(\frac{\partial F}{\partial x}\) and \(\frac{\partial F}{\partial y}\).
• Use these to compute slopes needed for tangent and normal lines.

Implicit differentiation helps navigate the complexity, making it easier to work with intricate curves and multi-variable relationships.