# Differantiation proof question

• transgalactic
In summary, the conversation is discussing limits and the concept of taking out a function from the limits. The speaker mentions how it doesn't matter whether the function is left in or not, as long as it can be considered a constant with respect to the limiting variable. They also provide an example of a limit where a function is taken out and discuss how the results can differ from the left and right. They then mention another function inside and how they are unsure of how to get a result. This leads to a discussion about the limit of |x|/x and its values from the left and right.
Well good, you got quite far. Look at where your limits got you to. You can take out the $$\phi (x)$$ out from the limits now can't you? Your remaining limit, what does that approach from the left? How about from the right?

Last edited:
i can't take out
$$\phi (x)$$

out of the limit .
its not a constant
??

Well really it doesn't matter whether you leave it in there or not, its still easy to see that we get different limits from the left and right just by considering the other things in the limit. But we can take it out of the limit because we can regard it as a constant with respect to the limiting variable.

i can't see
how we get different results

there is another function inside
i don't know how to get a result
??

Consider

$$\lim_{x\to 0} \frac{ |x| }{x}$$. What is it from the left? How about right? So it does exist?

## 1. What is differentiation proof?

Differentiation proof is a mathematical process used to find the rate of change of a function. It involves finding the derivative of a function, which represents the slope of the tangent line at any given point on the function.

## 2. How do you prove differentiation?

To prove differentiation, you must use the definition of the derivative, which states that the derivative of a function f(x) at a point x is equal to the limit as h approaches 0 of [f(x+h) - f(x)] / h. This limit will give you the slope of the tangent line at that point, also known as the derivative.

## 3. What is the difference between differentiation and integration?

Differentiation is the process of finding the derivative of a function, while integration is the process of finding the antiderivative of a function. In other words, differentiation finds the rate of change of a function, while integration finds the original function from its derivative.

## 4. How do you use differentiation in real life?

Differentiation has many real-life applications, including in physics, economics, and engineering. For example, it can be used to find the velocity and acceleration of an object at a given time, determine the optimal production level for a company, or calculate the slope of a curve in a circuit diagram.

## 5. Are there any shortcuts for differentiation?

Yes, there are several shortcut rules for differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules allow you to find the derivative of more complex functions without having to use the definition of the derivative every time. However, it is important to understand the definition and how these rules are derived in order to use them correctly.

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