# Differentiating with respect to $1 - x$

I am fairly sure this is a silly question, but a Google search was insufficient to find a satisfactory answer.

If I differentiate some function of $x$ with respect to $1-x$, what do I get compared to differentiating with respect to $x$?

I know I need to use the chain rule to figure this out, but I am stuck on the details.

• What does it mean to differentiate with respect to $1-x$??? – copper.hat Feb 7 '18 at 18:45
• @copper.hat I guess he considers $f(g(x))$ and wants to calculate $f^\prime(g(x))$ – Thomas Feb 7 '18 at 18:49
• @Thomas: I was hoping the OP might express what they want a little more clearly. – copper.hat Feb 7 '18 at 18:50
• Ok sorry, maybe it doesn't even make sense to do that. I was thinking that if I had some function of x, I could let u = 1 - x, and take the derivative with respect to u. The purpose of my question is that I learned that the deposit multiplier is the derivative of deposit size with respect to the reserve requirement. But my expression is MUCH easier to differentiate with respect to u = (1 - rr), if that is a thing I can do. – John Smith Feb 7 '18 at 18:51

If you mean $\frac{dy}{d(1-x)}$, that is $$\frac{dy}{d(1-x)} = \frac{dy}{dx} \cdot \frac{dx}{d(1-x)} = \frac{\frac{dy}{dx}}{\frac{d(1-x)}{dx}} = -\frac{dy}{dx}$$ because $\frac{d(1-x)}{dx} = -1$.
• That is gross. ${}$ – copper.hat Feb 7 '18 at 18:49
• That's what I understood from "differentiating with respect to $1-x$" as I indicated in the beginning. If OP doesn't mean that, I can simply edit this post according to what OP wants. – ArsenBerk Feb 7 '18 at 18:55
• What is the formal definition of the "symbol" $\frac{df}{dg}$? – Ixion Feb 7 '18 at 19:02
• a trick with notations is not a proof. The chain rule underneath doesn't justify any abuse of notation. So, go and learn what IS $df$. Good luck. – Netchaiev Feb 7 '18 at 19:14
If you have a function of $1-x$ and you want to differentiate that function with respect to $1-x$, start by setting $y:= 1 - x$. Then replace every $1-x$ with $y$ in the expression, and differentiate the expression with respect to the variable $y$. After differentiating, replace each of the $y$'s in the derivative with $1-x$.
For example, to differentiate $(1 - x)^{2}$ with respect to $1-x$, set $y := 1 - x$, and so the expression is now $y^{2}$. Differentiating this with respect to $y$ gives $2y$, and substituting $1-x$ back in gives the derivative as $2(1-x)$.