proved that the derivative of an odd functio is an even function
Answers
A function is even if f(−x) = f(x) for all x; similarly a function is odd if f(−x) = −f(x) for all x. Prove that
the derivative of an odd function is even, and that the derivative of an even function is odd. Hint: Problem 28 on
the textbook problems could prove useful.
We first prove that the derivative of an even function is odd. Start with the definition of f
0
(x):
f
0
(x) = lim
h→0
f(x + h) − f(x)
h
.
Now we want to show that f
0
(−x) = −f
0
(x) assuming that f(−x) = f(−x). So first we evaluate f
0
(−x):
f
0
(−x) = lim
h→0
f(−x + h) − f(−x)
h
.
We perform some algebraic manipulation:
lim
h→0
f(−x + h) − f(−x)
h
= lim
h→0
f(−(x − h)) − f(−x)
h
= lim
h→0
f(x − h) − f(x)
h
= −f
0
(x)
To get from line 1 to line 2 above, we use that fact that f(−x) = f(x). To get from line 2 to line 3 we use problem
28 from the textbook homework with c = −1.
The proof for the derivative of an odd function being even is similar. We wish to show that f
0
(−x) = f
0
(x)
assuming that f(−x) = −f(x).
f
0
(−x) = lim
h→0
f(−x + h) − f(−x)
h
= lim
h→0
f(−(x − h)) − f(−x)
h
= lim
h→0
−f(x − h) + f(x)
h
= lim
h→0
−
f(x − h) − f(x)
h
= − lim
h→0
f(x − h) − f(x)
h
= −(−f
0
(x))
= f
0
(x).
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