Understanding First Derivatives: Exploring Critical Points and Local Extrema

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In summary, The conversation discusses the topic of calculus and the use of derivatives. The person has been teaching themselves calculus and has a question about the introduction of x^2 - x - 2 in brackets in a derivative example. They mention using 'calculus for dummies' as a source of reading material and ask for clarification on what has happened in the example. The source of the conversation is from Harvey Mudd College and the response explains that factoring is being used to find critical points of the original function.
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PowerBuilder
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I don't post many questions here since I'm usually able to find most of my answers before starting a thread. As only a few of you may know I'm gradually teaching myself calculus. What can I say? I guess I'm that kind of guy.

I've managed to pick up on quite a few good basic ideas. There is one question I have about 'first derivatives'...

The derivative of ( 4x^3 ) is 12x^2. I understand that. How in the heck though does the below example work, in regards to the introduction of the x^2 - x - 2 in brackets?

I'm using 'calculus for dummies' as one source of reading material, besides the entire internet...but can't seem to find out...whats happened. I've missed a beat somewhere.

The function f(x)=3x^4 − 4x^3−12x^2+3 has first derivative

f(x) = 12x^3−12x^2−24x
= 12x (x^2 − x −2)
= 12x (x + 1)(x− 2)

Source - Harvey Mudd College
 
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  • #2
Just factor 12x out of each term
 
  • #3
That part is just algebra. They are factoring it algebraically to find critical points (where the first derivative is zero or undefined). As that page shows, it's useful for finding local maximum and minimum points of the original function.
 

FAQ: Understanding First Derivatives: Exploring Critical Points and Local Extrema

What is a first derivative?

A first derivative is a mathematical concept that represents the rate of change of a function at a specific point. It is also known as the slope or gradient of the function at that point.

How is a first derivative calculated?

A first derivative is calculated using the derivative formula, which is the limit of the difference quotient as the change in the independent variable approaches zero. This can also be written as the derivative of the function, f'(x), or dy/dx.

What is the significance of a first derivative?

The first derivative of a function can tell us important information about the behavior of the function, such as the direction of the function (increasing or decreasing), the location of maximum and minimum points, and the concavity of the function. It is also used to find the slope of a tangent line to a curve at a specific point.

What is the difference between a first derivative and a second derivative?

A first derivative represents the rate of change of a function, while a second derivative represents the rate of change of the first derivative. In other words, a second derivative is the derivative of a first derivative. It can also provide information about the curvature of a function.

How is a first derivative used in real-world applications?

First derivatives are commonly used in fields such as physics, engineering, and economics to model and analyze real-world phenomena. They can be used to calculate velocity, acceleration, and other rates of change. Additionally, they are used in optimization problems to find the maximum or minimum values of a function.

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