Navigation Panel: Previous | Up | Forward | Graphical Version | PostScript version | U of T Math Network Home

University of Toronto Mathematics Network
Question Corner and Discussion Area

Symmetry of Functions and their Derivatives

Asked by Sam Beroz on Monday Dec 11, 1995:
For every fuction with symmetry will the first derivative have symmetry of the other type? What Theorem proves this if it is in fact true?
Yes, it is true. If f is an even function (that is, has the same value if you replace x by -x), then its derivative will be an odd function (changes sign when you replace x by -x), and vice versa.

This is quite clear geometrically; in the picture below, for example, it is apparent that the slopes m and M are negatives of each other. You could even turn this into a geometric proof: if f is even, its graph is the same if you reflect it in a mirror placed along the y-axis, and therefore the tangent line at one point is the mirror reflection of the tangent line at the reflected point, and a line reflected in the y-axis has its slope multiplied by -1.

                           *        |        *
                slope m -->\*       |       */<-- slope M
                               *    |    *
                          -x                  x

In the above picture, m = -M.

The way that you prove it using only calculus theorems (without needing any geometry at all) is as follows.

If f is an even function, that means that f(x) = f(-x).

Now differentiate both sides. The left-hand side becomes f'(x), and the right-hand side becomes -f'(-x) (using the chain rule).

Therefore, f'(x) = - f'(-x). In other words, the value of f' at x is the negative of its value at -x, so f' is an odd function.

Similarly, if you started with an odd function f, you have f(x) = - f(-x). Differentiating both sides gives f'(x) = + f'(-x), so f' is an even function.

In either case, f' has the opposite type of symmetry (even or odd) from f.

[ Submit Your Own Question ] [ Create a Discussion Topic ]

This part of the site maintained by (No Current Maintainers)
Last updated: April 19, 1999
Original Web Site Creator / Mathematical Content Developer: Philip Spencer
Current Network Coordinator and Contact Person: Joel Chan -

Navigation Panel: 

  Go backward to Does Every Function Have an Antiderivative?
  Go up to Question Corner Index
  Go forward to Factorials of Non-Integral Values
  Switch to graphical version (better pictures & formulas)
  Access printed version in PostScript format (requires PostScript printer)
  Go to University of Toronto Mathematics Network Home Page