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The gradient of a function is a crucial concept when exploring directional derivatives. It represents how a function changes at any given point and directs us toward the steepest ascent. In two dimensions, for a function of two variables such as \( f(x, y) \), the gradient \( abla f \) is a vector consisting of the function's partial derivatives with respect to each variable.
The gradient is noted as:

• \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) \)

This vector tells us the direction of the most rapid increase in the function value. For the given function \( f(x, y) = xy + y^2 \), the gradient at any point \( (x, y) \) is calculated as follows:

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

Thus, the gradient of \( f \) is \( abla f = (y, x + 2y) \).Understanding gradients is the first step towards finding directions where derivatives will take specific values, including zero, which we explore using the directional derivative.

Partial derivatives are foundational when working with functions of multiple variables. They allow us to investigate how a function changes with respect to one variable while keeping others constant. In essence, partial derivatives are just derivatives in multi-variable calculus, focusing on one dimension of the problem at a time.
For the function \( f(x, y) = xy + y^2 \), its partial derivatives are:

• With respect to \( x \): \( \frac{\partial f}{\partial x} = y \)
• With respect to \( y \): \( \frac{\partial f}{\partial y} = x + 2y \)

These calculations show how the function changes as \( x \) or \( y \) varies independently. When evaluating at a specific point, like \( P(3,2) \), these derivatives provide the rate of change at that particular location.Calculating partial derivatives is key in determining the gradient, which in turn helps us solve problems related to directional derivatives and motion.

Understanding unit vectors is integral to calculating directional derivatives. A unit vector has a magnitude of 1 and is used to indicate direction. It plays a vital role in ensuring that when computing directional derivatives, we are analyzing shifts that are solely due to direction, not length.
A general unit vector \( \mathbf{u} \) in two dimensions is represented as \( (u_1, u_2) \), meeting the condition:

• \( u_1^2 + u_2^2 = 1 \)

When we solve for a zero directional derivative in Step 4, we need these unit vectors to comply with this condition. For our function's directional derivative \( D_{\mathbf{u}}f = abla f \cdot \mathbf{u} \), achieving zero means solving for vectors like \( \left(-\frac{7}{2}u_2, u_2\right) \).With careful calculation, solution vectors like \( \left(-\frac{7}{\sqrt{53}}, \frac{2}{\sqrt{53}}\right) \) emerge. These vectors are both unit vectors and represent directions where the rate of change, i.e., directional derivative, is precisely zero. Recognizing and using unit vectors is critical to solving directional derivative problems.