Derivatives of functions and equality of those functions

In summary, the question asks if two arbitrary real functions with equal derivatives imply that they are equal. However, this is not always the case, as shown by the example of f(x)=x^2+3 and g(x)=x^2-4. The correct statement is that they are equal up to an additive constant.
  • #1
Niles
1,866
0
Hi

I thought of something today coming home from school: If I have two arbitrary real functions f(x) and g(x) and I know that
[tex]
\frac{df(x)}{x} = \frac{dg(x)}{dx}
[/tex]
Does this imply that f(x)=g(x)?


Niles.
 
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  • #2


i think you df(x)/dx for the right hand side

no, they might not be the same.

As an example f(x)=x^2 +3 and g(x) = x^2 - 4

What do you get for df(x)/dx and dg(x)/dx?
 
  • #3


Yes, I meant dx, not x, in the denominator. Thanks for pointing that out.

OK, I see. So it must be correct up to an additive constant. Thanks.
 

1. What are derivatives of functions?

Derivatives of functions are mathematical tools used to measure the rate of change of a function at a specific point. They are essentially the slope of a function at a given point and can be used to analyze the behavior of functions.

2. How are derivatives of functions calculated?

There are several methods for calculating derivatives, but the most common is using the limit definition of a derivative. This involves taking the limit of the slope of a secant line as the two points on the function get closer and closer together.

3. What is the relationship between a function and its derivative?

The derivative of a function is essentially a new function that represents the slope of the original function at any given point. This means that the derivative can provide valuable information about the behavior and properties of the original function.

4. What is the chain rule in derivatives?

The chain rule is a rule used to find the derivative of a composite function, which is a function that is composed of two or more other functions. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

5. How can derivatives be used to find the maximum and minimum points of a function?

The derivative of a function can be used to find critical points, which are points where the derivative is equal to zero. The first derivative test can then be used to determine if these points are maximum or minimum points on the function. This can be helpful in optimization problems.

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