Partial derivatives using definition

In summary, the conversation discusses finding partial derivatives of a function f(x,y) around the point (2,3). The first partial derivatives f'x and f'y are easily found using the given definition. However, the conversation also mentions the need to find the second partial derivatives f''xx, f''yy, and f''xy, which can be found by applying the same method as before. The conversation also includes a discussion on the use of the definition and how to express it in terms of the given function.
  • #1
gamma-ray-burst
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Homework Statement


We are given a table where showing the points x and y and values of a function f(x,y).
The function itself is not given.
I have to find the partial derivatives f'x, f'y, f''xx, f''yy and f''xy around the point (2,3).


Homework Equations


I have to use the definition :
f'x(2,3) = lim (h-->0) [f(2+h,3) - f(2,3)] / h


The Attempt at a Solution


Ok. So I easily found f'x and f'y. (h is found by looking at the xs and ys, h for f'x = -1 and 2 for f'y).
My problem is finding the f''. My teacher is the kind that shows one definition and gives one example. I don't know how to find the f''s.

If someone could just point the way, I'd be grateful.
Thanks!
 
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  • #2
The definition of the second partials is just the partial derivatives of the first partials. Why couldn't you just use the same method as before?
 
  • #3
Well I know. It's easy to find the partial derivatives when we have the function. That I can find.
But as I said, we have just done one example with the "definition", so I quite don't see how.

It's like d[df/dx]/dx
When you derivate the function you just assume y as constant.
But when it comes to lim f(a+h,b)-f(a,b)/h, I just don't know how to express it.

f'x = 0 and f'y = 2
f''xx(2,3) = f'x [f(2+h, 3) - f(2,3) /h]
= f(2+2h, 3) - f(2,3)/h ?

I don't know really.
And then again, I'm quite stupid :D
 

1. What is the definition of a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant.

2. How do you calculate a partial derivative using the definition?

To calculate a partial derivative using the definition, you take the limit of the difference quotient as the variable approaches a specific value. The difference quotient is calculated by dividing the change in the function's output by the change in the variable's input.

3. What is the purpose of using partial derivatives in science?

Partial derivatives are used in science to better understand how a function behaves when only one of its variables is changing. This can be helpful in fields such as physics, engineering, and economics, where multiple variables are often involved in complex systems.

4. What does it mean to take a partial derivative with respect to a specific variable?

When taking a partial derivative with respect to a specific variable, you are essentially asking "how does the output of this function change as this variable changes, while keeping all other variables constant?" This allows us to isolate and analyze the impact of a single variable on a function.

5. Can you provide an example of calculating a partial derivative using the definition?

Sure, let's say we have a function f(x,y) = 2x + 3y. To calculate the partial derivative with respect to x, we would take the limit as h approaches 0 of [f(x+h,y) - f(x,y)] / h. This would result in the derivative 2, since the y term drops out when taking the limit. Similarly, to find the partial derivative with respect to y, we would take the limit as h approaches 0 of [f(x,y+h) - f(x,y)] / h, which also results in a derivative of 3.

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