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

[DavidLiew]

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

 

[quasar987]

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

 

[ross_tang]

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]

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.

 

[ross_tang]

quasar987, negative logarithm does indeed exist.

http://en.wikipedia.org/wiki/Logarithm#Logarithm_of_a_negative_or_complex_number"

But I think you are right to assume x is positive.

 

[quasar987]

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

But thanks for the reminder.