Is It Possible to Have Bounded Partial Derivatives Without Differentiability?

In summary, differentiability and boundedness of partial derivatives are not necessarily related. An example of a function with bounded partial derivatives but not differentiable is f(x) = |x|.
  • #1
Diffy
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Is it possible to have a function that has bounded paritial derivatives, but is not differential? Can you give me an example? And if possible explain how this is possible?

I am having trouble understanding this calculus concept. Thanks.
 
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  • #2
I assume that you meant "differentiable" instead of "differential"?

How about f(x) = |x|.
For all x, [tex]|f'(x)| \le 1[/tex]
however, f(x) is not differentiable.

The boundedness of partial derivatives and differentiability don't have much to do with each other. On the other hand, for example, the derivative of f(x) = x2 is unbounded although f(x) is perfectly well differentiable (infinitely often, even).
 

1. What are partial derivatives?

Partial derivatives are a type of derivative that measures the rate of change of a function with respect to one of its variables while holding all other variables constant. They are used in multivariate calculus to analyze functions with multiple variables.

2. How are partial derivatives calculated?

Partial derivatives are calculated by taking the derivative of a function with respect to one variable while treating all other variables as constants. This can be done using the chain rule and basic derivative rules.

3. What is the difference between partial derivatives and ordinary derivatives?

The main difference between partial derivatives and ordinary derivatives is that partial derivatives measure the rate of change of a function with respect to one variable at a specific point, while ordinary derivatives measure the overall rate of change of a function along its entire curve.

4. What are some real-world applications of partial derivatives?

Partial derivatives have many real-world applications, including in physics, economics, and engineering. They are used to model and analyze multivariate systems, such as the forces acting on a moving object, the relationship between supply and demand in a market, or the optimal design of a structure.

5. Can partial derivatives be used to find maximum or minimum values?

Yes, partial derivatives can be used to find maximum or minimum values of a function with multiple variables. This is done by setting the partial derivatives equal to zero and solving for the variables, which gives critical points that can be evaluated to determine the maximum or minimum values.

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