How do you find the derivative of the inverse function of: x/(x-5)?

Question

agent0smith · Answer

Firstly, do you know how to find the inverse?

anonymous · Answer

By switching xs and ys.

agent0smith · Answer

Yes... but i meant CAN you find the inverse?

anonymous · Answer

I've sorta forgotten how to...

agent0smith · Answer

$$\Large y = \frac{ x }{ x-5}$$multiply both sides by x-5$$\Large y(x-5) = \frac{ x }{ x-5}*(x-5)$$which simplifies to...

anonymous · Answer

x=yx-5y?

agent0smith · Answer

Correct. Now you need to get all the x's on one side of the equation (the -5y can stay where it is)

anonymous · Answer

Okay. Would I divide the x from both sides?

agent0smith · Answer

No, just subtract yx from both sides... then we'll be able to factor out an x on the left side.

anonymous · Answer

What happens to the y?

anonymous · Answer

x(-y+1)=-5y is this correct?

agent0smith · Answer

Wow... Awesome! you already factored it too! Now... see how you can get x alone?

anonymous · Answer

Yep. But, the y...

agent0smith · Answer

haha don't worry about the y... remember, once you get x=... you swap x and y :)

anonymous · Answer

Oh! Hmmm. I was feel like I was taught differently. Thanks! Now to find the derivative, I'd use the Product Rule, multiply it by the original and... I'm confused.

agent0smith · Answer

Well you can swap x and y before you start finding the inverse, if that's what you meant?

agent0smith · Answer

Okay so anyway you should have $$\Large y = \frac{ -5x }{ 1-x }$$

Know how to use the quotient rule?

anonymous · Answer

Right, that's what I got. And yes.

agent0smith · Answer

Okay... want to try using it? I'll help if you get stuck.

anonymous · Answer

Would the derivative of g(x) be zero? f(x)=5x and g(x)=x-1, right?

agent0smith · Answer

damn you're fast :D i'm guessing you already have it like this $$\Large y = \frac{ 5x }{ x -1}$$
but no, derivative of x-1 is still 1. Derivative of 5x is 5.

anonymous · Answer

I got: $$\frac{ -5 }{ (x-2)^{2} }$$

agent0smith · Answer

Why'd the denominator become x-2? It shouldn't have changed other than the squared. lemme check the rest

agent0smith · Answer

otherwise you're right, very nice job http://www.wolframalpha.com/input/?i=derivative+y+%3D+%5Cfrac%7B+5x+%7D%7B+x+-1%7D

anonymous · Answer

Sorry. Yes, the two should have been a one. You're awesome. Thanks!!!

anonymous · Answer

^^

agent0smith · Answer

Good job! :D