91Ó°ÊÓ

Partial derivatives are a key component in multivariable calculus, especially when we have functions with more than one variable. A partial derivative measures how a function changes as one of its variables changes, keeping other variables constant.
For example, if we have a function of two variables, like the function given in the problem:
\[f(x, y) = 4x^2 - 2xy + y^2\]
we can find its partial derivatives with respect to each variable.

• To find the partial derivative with respect to \(x\), symbolized as \(\frac{\partial f}{\partial x}\), treat \(y\) as a constant. Thus, \[\frac{\partial f}{\partial x} = 8x - 2y\].
• Similarly, for the partial derivative with respect to \(y\), represented as \(\frac{\partial f}{\partial y}\), treat \(x\) as a constant to arrive at \[\frac{\partial f}{\partial y} = -2x + 2y\].

Partial derivatives are foundational for constructing other concepts like tangent planes and, pivotal in our discussion, the gradient vector.

The gradient vector is an essential concept in understanding how multi-variable functions behave. It provides a vector that points in the direction of the steepest increase of a function. Formally, it's the vector composed of all the partial derivatives of the function.
For our function \(f(x, y) = 4x^2 - 2xy + y^2\), the gradient vector is:
\[abla f(x, y) = \left\langle 8x - 2y, -2x + 2y \right\rangle\]
Here is what this expression tells us:

• The "8x - 2y" component represents the rate of change of the function regarding the \(x\)-axis.
• Likewise, the "-2x + 2y" component represents how the function varies along the \(y\)-axis.

The gradient vector essentially gives us the direction for the fastest way to "climb the hill" of the function at any point \((x, y)\). Its magnitude tells us how steep that climb is. Understanding this helps in many applications, such as in optimization problems where you want to find minima or maxima of functions.

Evaluating a gradient vector at a specific point allows us to find the exact rate of increase of a function at that particular location. For our problem, we are given the point \(P(-1, -5)\), and we need to determine how the function behaves there.
To do so, you substitute the point's coordinates into the gradient vector:
\[abla f(-1, -5) = \left\langle 8(-1) - 2(-5), -2(-1) + 2(-5) \right\rangle\]
This simplifies to:

• \(8(-1) - 2(-5) = -8 + 10 = 2\)
• \(-2(-1) + 2(-5) = 2 - 10 = -8\)

Thus, the gradient vector at \((-1, -5)\) is \(\left\langle 2, -8 \right\rangle\).
This means at point \((-1, -5)\), the function \(f\) increases fastest in the direction \(\left\langle 2, -8 \right\rangle\). This vector also gives insight into the slope or steepness—indicating directions for maximum function increase.