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

## Main Question or Discussion Point

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

quasar987
Homework Helper
Gold Member

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

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:

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

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

quasar987
Homework Helper
Gold Member

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.

quasar987