# How to differentiate y = 8 ln x - 9 x with respect to x^2?

1. Aug 17, 2010

### DavidLiew

How to differentiate y = 8 ln x - 9 x with respect to x^2?

2. Aug 17, 2010

### quasar987

Re: differentiation

Introduce the new variable z=x². Then x=√z and y(z)=8ln√z - 9√z. Now differentiate wrt z.

3. Aug 17, 2010

### ross_tang

Re: differentiation

Hello, quasar. I don't think using square root function is a good method, since you don't know whether to take the positive or negative square root. For more difficult case, you may not be able to find the inverse function.

To differentiate f(x) w.r.t g(x), just do the following:

$$\frac{d f(x)}{d g(x)}$$

$$=\frac{d f(x)}{d x}\frac{d x}{d g(x)}$$

$$=\frac{d f(x)}{d x}\left(\frac{d g(x)}{d x}\right)^{-1}$$

David, for your problem, you just put:

$$f(x)=8 \ln x -9x$$ and $$g(x) = x^2$$, and you will have the answer:

$$\frac{\frac{8}{x}-9}{2x}$$

$$=\frac{8-9 x}{2 x^2}$$

4. Aug 17, 2010

### quasar987

Re: differentiation

ross_tang, x is always positive in this problem because of the "ln x" appearing in the defining formula for y. So I don't think there is an ambiguity of any sort.

5. Aug 17, 2010

### ross_tang

Last edited by a moderator: Apr 25, 2017
6. Aug 17, 2010

### quasar987

Re: differentiation

Prrrrrety sure the domain of the function the OP is interested in is a subset of ]0,infty[. :)

But thanks for the reminder.