Math Problem Statement

explain simplify and proove with examplea

Explain, simplify, and prove partial derivatives and gradients with examples.

Solution

Let's go through the concepts of partial derivatives and gradients as presented in your images.

Partial Derivatives (Image 1)

Definition:

Let f:DRnRf: D \subseteq \mathbb{R}^n \rightarrow \mathbb{R} be a function, where a=(a1,a2,,an)URna = (a_1, a_2, \dots, a_n) \in U \subseteq \mathbb{R}^n is a point. The partial derivative of ff at the point aa with respect to the variable xix_i, denoted fxi(a)\frac{\partial f}{\partial x_i}(a), is the rate of change of ff in the direction of the ii-th coordinate axis.

Formula:

The first partial derivative of ff in the direction of the vector eie_i is defined by: fxi(a)=Deif(a)=limt0f(a+tei)f(a)t\frac{\partial f}{\partial x_i}(a) = D_{e_i}f(a) = \lim_{t \to 0} \frac{f(a + t e_i) - f(a)}{t} Where:

  • eie_i is the unit vector in the direction of the xix_i-axis.
  • a+teia + t e_i represents moving in the xix_i-direction by a small amount tt.

This can be expanded as: fxi(a)=limxiaif(a1,a2,,xi,,an)f(a1,a2,,ai,,an)xiai\frac{\partial f}{\partial x_i}(a) = \lim_{x_i \to a_i} \frac{f(a_1, a_2, \dots, x_i, \dots, a_n) - f(a_1, a_2, \dots, a_i, \dots, a_n)}{x_i - a_i} Thus, the partial derivative measures how the function changes as xix_i varies, keeping all other variables constant.

Example:

Suppose f(x,y)=x2+y2f(x, y) = x^2 + y^2. The partial derivatives are:

  • fx=2x\frac{\partial f}{\partial x} = 2x
  • fy=2y\frac{\partial f}{\partial y} = 2y

At the point (x0,y0)=(1,2)(x_0, y_0) = (1, 2):

  • fx(1,2)=2(1)=2\frac{\partial f}{\partial x}(1, 2) = 2(1) = 2
  • fy(1,2)=2(2)=4\frac{\partial f}{\partial y}(1, 2) = 2(2) = 4

Gradient (Image 2)

Definition:

The gradient of ff at the point aRna \in \mathbb{R}^n is a vector that consists of all the partial derivatives of ff. It is denoted by f(a)\nabla f(a) or grad f(a)\text{grad} \ f(a).

Formula:

The gradient of ff is the vector: f(a)=(fx1(a),,fxn(a))\nabla f(a) = \left( \frac{\partial f}{\partial x_1}(a), \dots, \frac{\partial f}{\partial x_n}(a) \right) This vector points in the direction of the steepest ascent of the function ff and its magnitude represents the rate of the steepest increase.

Example:

Using the same function f(x,y)=x2+y2f(x, y) = x^2 + y^2, the gradient at (1,2)(1, 2) is: f(1,2)=(fx(1,2),fy(1,2))=(2,4)\nabla f(1, 2) = \left( \frac{\partial f}{\partial x}(1, 2), \frac{\partial f}{\partial y}(1, 2) \right) = (2, 4) This means that the function increases fastest in the direction of the vector (2,4)(2, 4) at the point (1,2)(1, 2).

Simplified Summary:

  1. Partial derivatives give the rate of change of a function along one direction (keeping others constant).
  2. The gradient is a vector of all partial derivatives, giving the direction and rate of the steepest increase in the function.
  3. Partial derivatives are used to form the gradient vector.

Would you like more details, or have any specific questions?

Related Questions:

  1. How can partial derivatives be used to find local maxima and minima of a function?
  2. What is the geometric interpretation of a gradient?
  3. Can you explain how partial derivatives apply in multivariable optimization?
  4. How does the gradient relate to directional derivatives?
  5. What are higher-order partial derivatives, and when are they used?

Tip:

When working with gradients, remember that they always point in the direction of the steepest ascent!

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Partial Derivatives
Gradient

Formulas

Partial Derivative: ∂f/∂xi(a) = lim(t→0) [f(a + tei) - f(a)] / t
Gradient: ∇f(a) = (∂f/∂x1(a), ..., ∂f/∂xn(a))

Theorems

Definition of Partial Derivatives
Gradient of a Multivariable Function

Suitable Grade Level

College Level (Advanced Calculus)