Use of a derivative or a gradient to minimize a function

In summary, there are two main methods for minimizing a function: setting its derivative to zero and solving for x, or using the method of gradient descent which takes multiple steps. The reason for using gradient descent is when the function cannot be differentiated analytically. An example of this is when the data is given as a set of pairs from an experiment rather than an analytic function.
  • #1
onako
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Given certain function f(x), a standard way to minimize it is to set its derivative to zero, and solve for x. However, in certain cases the method of gradient descent is used; compared to the previous method (call it 'method I')that simply sets the derivative to zero and solves for x, the gradient descent takes multiple steps.

Why could not one use only the 'method I' for minimization? Could you give an example illustrating the difficulty of applying 'mehtod I'?
 
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  • #2
The standard situation is where you cannot differentiate the function analytically and have to use a numerical approximation.
 
  • #3
Could you provide a simple example?
 
  • #4
Any situation in which your data is given as a set of pairs from an experient rather than as an analytic function.
 

1. What is a derivative?

A derivative is a mathematical tool that represents the rate of change of a function at any given point. It tells us how a function is changing, or the slope of the curve, at a specific point.

2. How is a derivative used to minimize a function?

A derivative is used to minimize a function by finding the point where the slope of the function is equal to 0. This point, known as the critical point, is where the function reaches its minimum value. By setting the derivative to 0 and solving for the variable, we can find this critical point and use it to minimize the function.

3. What is a gradient?

A gradient is a vector that represents the direction and magnitude of the steepest ascent or descent of a function at any given point. It is a generalization of the derivative for functions with multiple variables.

4. How is a gradient used to minimize a function?

A gradient is used to minimize a function by following its direction of steepest descent. This means taking small steps in the opposite direction of the gradient until the function reaches its minimum value. This method is known as gradient descent.

5. What are some real-world applications of using a derivative or gradient to minimize a function?

The use of derivatives and gradients to minimize functions has many practical applications in fields such as economics, engineering, and physics. For example, it can be used to optimize production processes, design efficient structures, and model complex systems. It is also widely used in machine learning algorithms to find the optimal values for parameters in predictive models.

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