How do you differentiate a function with respect to the negative of its variable. For example, is it true that df(-x)/dx = - df(x)/dx? If so, why is it?
Asked
Active
Viewed 2,771 times
1 Answers
3
It is confusing because you are using $x$ in two ways: as the argument of $f$ and as the variable you vary. $f(x)$ and $f(-x)$ refer to two different arguments of $f$, and the values may have nothing to do with each other. For example, take $f(x)=x$. Then $f(-x)=-x$ and $\frac {df(-x)}{dx}=-1=-\frac {df(x)}{dx}$ On the other hand,take $f(x)=x^2$, so $f(-x)=f(x)$ Then the derivatives are equal, not negatives. What is true is $\frac {df(x)}{d(-x)}=-\frac {df(x)}{dx}$, but this is not the same thing.
Ross Millikan
- 374,822
-
-
1That follows from the chain rule. Define $u=-x$ and you want $\frac {df(x)}{du}$ – Ross Millikan Feb 24 '14 at 04:35