Differentiation Help: Solving x|x| & g(x) Problems

In summary: Other than that, both examples seem to require some algebraic manipulations and maybe some knowledge of the properties of differentiable functions.In summary, the conversation is about determining where the function x|x| is differentiable and finding its derivative. The participants discuss different approaches, such as using the product rule and the epsilon delta method, and ask for clues on how to solve these types of problems. They also discuss another problem involving a piecewise-defined function and how to show its differentiability. They suggest using difference quotients or analyzing the function graphically to determine differentiability, and also mention using algebraic manipulations and knowledge of differentiable functions.
  • #1
semidevil
157
2
i'm looking at this, but don't really know how to approach this.

determine where the function x|x| is differentiable and find the derivative.

i'm looking at this, but just have no idea where to start? I mean, I know x is differentiable everywhere, and |x| is not differentable at 0. looking at this, I'm thinking about the product rule, and I'm tryint to see if I need to solve this using epsilon delta method, but have no idea.

any clues just on how to solve these types of problems?

and another one:

g(x) = x^2/xin(1/x^2) for x != 0 and g(0) = 0. I'm suppose to show that g is differntaible everywhere.

how do I show these types of problems??
 
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  • #2
For x not=0, there is no problem with either of your examples. To handle the case of x=0, you could try forming difference quotients and letting the increment go to 0. For both examples you have to work with two difference quotients, positive increment and negative increment. If they both have the same limit as the increment goes to 0, then the derivative exists.
 
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  • #3
Another way is to analyze the function graphically. If there are any discontinuities or "sharp turns" where the limits on either side are not the same, then the function is not differentiable there.
 

FAQ: Differentiation Help: Solving x|x| & g(x) Problems

What is differentiation and why is it important?

Differentiation is a mathematical process used to find the rate of change or slope of a function at a specific point. It is important because it allows us to analyze and understand the behavior of a function and its relationship to other variables.

How do I differentiate a function with absolute value?

To differentiate a function with absolute value, you can use the chain rule. First, rewrite the function as two separate pieces: one for the positive value and one for the negative value. Then, take the derivative of each piece and use the chain rule to combine them back together.

What is the difference between x|x| and g(x)?

x|x| and g(x) are two different types of functions. x|x| is a piecewise function, meaning it has different rules for different parts of the domain. g(x) is a general function that can be differentiated using standard differentiation rules. The main difference is that x|x| may require the use of the chain rule, while g(x) can be differentiated using basic rules such as the power rule or product rule.

How do I know which differentiation rule to use?

The rule you use for differentiation depends on the type of function you are working with. For example, if you have a polynomial function, you can use the power rule. If you have a trigonometric function, you can use the trigonometric identities. It is important to practice and familiarize yourself with the different rules to know which one to use in each situation.

Can differentiation help me solve real-world problems?

Yes, differentiation can be used to solve a variety of real-world problems, such as finding the maximum or minimum value of a function, determining the velocity or acceleration of an object, and optimizing a process. Many fields, including physics, economics, and engineering, use differentiation to model and solve practical problems.

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