Derivative of X^2

[MrAl]

Hi,

To add to those questions, my question would be why would we have to resort to three or more dimensions in order to define the derivative. The derivative is easily defined in just two and will actually be required in many cases. For example, what is the tangent plane to y=x^2+1 at x=3 ? There would be multiples, but since we are working in just two dimensions we cant really do it anyway.

Just for reference, the formula df(x)/dx=(f(x+h)-f(x-h))/(h+h) is sometimes called the central means derivative. It has higher numerical accuracy than the usual definition.

Also just for reference, the derivative of abs(x) at x=0 is sometimes taken to be 0.

 

[Spuriosity]

How can the above equation be a definition of f '(x) when it includes f '(a) ? What is the definition of f '(a)? Isn't that a circular definition?

I see what you mean, but that equation only involves f(x) (undashed - that's not a typo) f(a) and xi(x), all of which are known in the limit as x-> a. This is the definition of f'(x) evaluated at the point x=a. (if this implicit form makes you uneasy, you can rearrange the equation as well.

How can the sum of x plus another quantity be equal to itself?

Sorry, that was just lazy/bad notation. I meant to say that we evaluate everything at x+a instead of x, but I've changed it to h for clarity.

What is the definition of xi(x+a)?

xi(x+a) is defined as the difference between the tangent line and the function itself. (see diagram):
**broken link removed**

... my question would be why would we have to resort to three or more dimensions in order to define the derivative. The derivative is easily defined in just two and will actually be required in many cases. For example, what is the tangent plane to y=x^2+1 at x=3 ? There would be multiples, but since we are working in just two dimensions we cant really do it anyway.

Sorry for the confusion - When I said 'tangent plane', it would've been better to say 'tangent line'. All of the working was for a 1D derivative (it's only one-dimensional since it's f(x) only - a 2D derivative would require f(x,y) ). the weird quantities like a and h are just 1D displacements along the x axis, while xi is just a different function of x. No higher dimensions are involved.