Math Problem Statement
explain simplify and proove with examplea
Solution
Let's go through the concepts of partial derivatives and gradients as presented in your images.
Partial Derivatives (Image 1)
Definition:
Let be a function, where is a point. The partial derivative of at the point with respect to the variable , denoted , is the rate of change of in the direction of the -th coordinate axis.
Formula:
The first partial derivative of in the direction of the vector is defined by: Where:
- is the unit vector in the direction of the -axis.
- represents moving in the -direction by a small amount .
This can be expanded as: Thus, the partial derivative measures how the function changes as varies, keeping all other variables constant.
Example:
Suppose . The partial derivatives are:
At the point :
Gradient (Image 2)
Definition:
The gradient of at the point is a vector that consists of all the partial derivatives of . It is denoted by or .
Formula:
The gradient of is the vector: This vector points in the direction of the steepest ascent of the function and its magnitude represents the rate of the steepest increase.
Example:
Using the same function , the gradient at is: This means that the function increases fastest in the direction of the vector at the point .
Simplified Summary:
- Partial derivatives give the rate of change of a function along one direction (keeping others constant).
- The gradient is a vector of all partial derivatives, giving the direction and rate of the steepest increase in the function.
- Partial derivatives are used to form the gradient vector.
Would you like more details, or have any specific questions?
Related Questions:
- How can partial derivatives be used to find local maxima and minima of a function?
- What is the geometric interpretation of a gradient?
- Can you explain how partial derivatives apply in multivariable optimization?
- How does the gradient relate to directional derivatives?
- 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)
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