Functions with increasing derivatives

In summary, there are many functions, such as f(x) = e^(ax) where a > 1 and f(x) = 0 + x + 2x^2 + 3x^3 + ..., that have the property where the nth derivative evaluated at a point is less than the n+1th derivative evaluated at that point. These functions can also be infinite power series or simple functions like f(x) = -1/x for -1 < x < 0. The proof and convergence of these functions are left as exercises for the original poster.
  • #1
1MileCrash
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Consider a function f(x), such that for all points x0 in the domain, the nth derivative of f evaluated that x0 is less than the n+1th derivative of f evaluated at x0.

A quick example is f(x) = e^(ax) where a > 1, what others are there (not including just changing e to something else)? Is there a name for these?
 
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  • #2
It's easy to invent infinite power series that have this property. For example ##f(x) = 0 + x + 2x^2 + 3x^3 + \dots##.

Proving (a) it has the required property and (b) it is convergent (for some real values of ##x##) are left as exercises for the OP.

Actually, you don't need the infinite series. ##f(x) = -1/x##, when ##-1 < x < 0##.
 
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FAQ: Functions with increasing derivatives

What is the definition of a function with increasing derivatives?

A function with increasing derivatives is a mathematical function where the value of the derivative at any point on the function is greater than the value of the derivative at any point before it.

How do you determine if a function has increasing derivatives?

To determine if a function has increasing derivatives, you can take the derivative of the function and check if the resulting function is always positive. If it is, then the original function has increasing derivatives.

What is the significance of a function with increasing derivatives?

A function with increasing derivatives indicates that the rate of change of the function is increasing, meaning the function is becoming steeper as the input increases. This can be useful in analyzing the behavior of a system or predicting future values of the function.

Can a function have increasing derivatives at every point?

Yes, a function can have increasing derivatives at every point. This type of function is called a strictly increasing function, and it means that the function is always getting steeper as the input increases.

Are there any real-world examples of functions with increasing derivatives?

Yes, there are many real-world examples of functions with increasing derivatives. One example is the position-time graph of a car accelerating uniformly. The derivative of the position-time graph, which represents the car's velocity, will be constantly increasing as the car accelerates.

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