Small question about derivatives and gradients.

In summary, a derivative is a mathematical concept that represents the rate of change of a function at a particular point, and it is essential for analyzing the behavior of functions in various fields. Derivatives can be both positive and negative, and a gradient is a vector that represents the steepest slope of a function at a point, which is a generalization of the derivative for multivariable functions. Derivatives and gradients are closely related, with the gradient being a vector of all the function's derivatives in different directions.
  • #1
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Is the grad([itex]\frac{\partial f}{\partial t}[/itex]) the same as [itex]\frac{\partial}{\partial t}[/itex](gradf)?

Thank you.
 
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  • #2
Hi M. next! :wink:
M. next said:
Is the grad([itex]\frac{\partial f}{\partial t}[/itex]) the same as [itex]\frac{\partial}{\partial t}[/itex](gradf)?

Unless f is really weird, yes. :smile:

(it's the same test as whether ∂2f/∂x∂t = ∂2f/∂t∂x …

i forget the exact conditions)​
 
  • #3
Thank you!
 

1. What is a derivative?

A derivative is a mathematical concept that represents the rate of change of a function at a particular point. It is essentially the slope of a tangent line to the function at that point.

2. Why are derivatives important?

Derivatives are important because they allow us to analyze the behavior of a function, such as its rate of change, concavity, and extrema. They are also crucial in many areas of science, engineering, and economics, where they are used to model and optimize systems.

3. Can derivatives be negative?

Yes, derivatives can be negative. A negative derivative indicates that the function is decreasing at that point, while a positive derivative indicates that the function is increasing.

4. What is a gradient?

A gradient is a vector that represents the direction and magnitude of the steepest slope of a function at a particular point. It is a generalization of the derivative for multivariable functions.

5. How are derivatives and gradients related?

Derivatives and gradients are closely related, as the gradient of a function is the vector of its partial derivatives with respect to each input variable. In other words, the gradient is the vector of all the function's derivatives in different directions.

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