91Ó°ÊÓ

The gradient is a fundamental concept when dealing with functions of several variables. It provides a vector of the function's partial derivatives.
The gradient points in the direction of the steepest ascent of a function. It is represented as \( abla f \), where \( f \) is the function in consideration.
In the exercise, we have the function \( f(x, y) = y^{10} \). The gradient would then be the combination of partial derivatives with respect to each variable.

• For \( x \), the partial derivative is \( \frac{\partial f}{\partial x} \). Since \( f(x, y) \) does not include \( x \), this derivative is 0.
• For \( y \), the partial derivative is \( \frac{\partial f}{\partial y} \), calculated as \( 10y^9 \) using the power rule.

Thus, the gradient is \( abla f = (0, 10y^9) \). This gradient becomes a powerful tool for identifying the behavior of \( f(x, y) \) at different points.

Partial derivatives highlight how a function changes as only one variable changes, keeping others constant. This technique is pivotal when analyzing functions with several variables, such as \( f(x, y) \).
To find partial derivatives:

• Select the variable for differentiation, treating all others as constants.
• Differentiate the function with respect to this chosen variable.

In our exercise:

• For \( \frac{\partial f}{\partial x} \), as there is no \( x \) in the function \( f(x, y) = y^{10} \), the derivative is simply 0.
• For \( \frac{\partial f}{\partial y} \), applying the power rule results in \( 10y^9 \).

Partial derivatives are essential for forming the gradient vector \( abla f = (0, 10y^9) \), which supplies key data about \( f(x, y) \) with respect to its variables.

The dot product is a mathematical operation that takes two vectors and returns a scalar.
It provides insights into the relationship between the directions of these vectors, often used to project one vector along the direction of another.
Given vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), the dot product is:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]In this exercise, the directional derivative is computed as the dot product of the gradient \( abla f = (0, -10) \) at point \( P \) and the direction vector \( \mathbf{u} = (0, -1) \):

• The dot product calculation is: \( 0 \times 0 + (-10) \times (-1) = 10 \).

This calculation leads us to the directional derivative, illustrating how much the function increases in the direction of \( \mathbf{u} \).

The power rule is a basic yet powerful tool for differentiating functions, especially when dealing with polynomial terms.
For a function \( f(x) = x^n \), the derivative is \( f'(x) = nx^{n-1} \).
This rule simplifies calculus operations by providing a straightforward approach to derivative calculations.
In the context of our exercise, the function \( f(x, y) = y^{10} \) requires using the power rule for its partial derivative with respect to \( y \):

• Applying the power rule, \( \frac{\partial f}{\partial y} = 10y^{9} \).

The rule helps form part of the gradient, \( abla f = (0, 10y^9) \), dramatically simplifying the derivation process and facilitating further operations like computing directional derivatives.