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The gradient is a powerful concept in multivariable calculus. It's essentially a vector that contains all the partial derivatives of a function. In simpler terms, think of the gradient as a list of all the ways a function can change in every direction at a specific point.
For a function of two variables, like our function here, \(f(x, y) = 2xy - 3y^2\), the gradient is a vector comprising its partial derivatives with respect to each variable.

• Partial Derivative with respect to \(x\): It tells how the function changes as we vary \(x\) while keeping \(y\) constant. In our example, it is \(\frac{\partial f}{\partial x} = 2y\).
• Partial Derivative with respect to \(y\): Similarly, this shows how the function changes as \(y\) changes while \(x\) remains fixed. Here, it is \(\frac{\partial f}{\partial y} = 2x - 6y\).

The gradient vector for our function is \(abla f = (2y, 2x - 6y)\). By plugging in specific values of \(x\) and \(y\), as in the point \(P_0(5, 5)\), we can evaluate the gradient at that point to find \(abla f(5, 5) = (10, -20)\). This tells us the direction and rate of fastest increase of the function at that point.

Partial derivatives are a fundamental tool in calculus, especially when dealing with functions of multiple variables. They represent the derivative of a function with respect to just one of those variables, treating all the others as constants.
In the most intuitive sense, a partial derivative helps us understand how the function behaves as we tweak one variable while keeping others fixed.

• For our function \(f(x, y)\), the partial derivative with respect to \(x\), \(\frac{\partial f}{\partial x} = 2y\), shows how \(f\) changes with small changes in \(x\).
• The partial derivative with respect to \(y\), \(\frac{\partial f}{\partial y} = 2x - 6y\), similarly reveals how \(f\) changes when \(y\) is slightly altered.

Understanding these derivatives allows us to construct the gradient, which combines these paths of change into a single vector. Each component of this gradient vector is essentially a partial derivative. This knowledge is pivotal when we want to find how the function behaves in specific directions.

A unit vector plays a crucial role when we need to explore how a function changes in a particular direction.
Unlike any random vector, a unit vector has a magnitude of exactly one, ensuring that it only affects the direction without scaling the result. To find a unit vector, you take any non-zero vector and divide it by its magnitude.

• In our problem, the direction \(\mathbf{u} = 4\mathbf{i} + 3\mathbf{j}\) is a vector that points in a specific direction.
• Its magnitude, or length, is calculated using the formula: \(||\mathbf{u}|| = \sqrt{4^2 + 3^2} = 5\).
• Thus, the unit vector, \(\hat{\mathbf{u}}\), is derived by dividing \(\mathbf{u}\) by its magnitude: \(\left(\frac{4}{5}, \frac{3}{5}\right)\).

This unit vector now represents the direction without changing the size of how much we consider moving in that direction. When we compute the dot product of the gradient and this unit vector, we're calculating the directional derivative. This operation tells us how the function shifts in that specific direction, scaled appropriately.