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

In summary, the conversation discusses different methods for differentiating a function with respect to x^2. The suggested method involves introducing a new variable z and using the chain rule to differentiate. However, there is a concern about using the square root function and the ambiguity it can create. The alternative method involves using the quotient rule and setting f(x) = 8 ln x - 9x and g(x) = x^2. It is also noted that the function in question has a domain of ]0, infinity[, so there is no ambiguity in the use of the natural logarithm.
  • #1
DavidLiew
16
0
How to differentiate y = 8 ln x - 9 x with respect to x^2?
 
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  • #2


Introduce the new variable z=x². Then x=√z and y(z)=8ln√z - 9√z. Now differentiate wrt z.
 
  • #3


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:

[tex]\frac{d f(x)}{d g(x)}[/tex]

[tex]=\frac{d f(x)}{d x}\frac{d x}{d g(x)}[/tex]

[tex]=\frac{d f(x)}{d x}\left(\frac{d g(x)}{d x}\right)^{-1}[/tex]

David, for your problem, you just put:

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

[tex]\frac{\frac{8}{x}-9}{2x}[/tex]

[tex]=\frac{8-9 x}{2 x^2}[/tex]
 
  • #4


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.
 
  • #6


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

But thanks for the reminder.
 

1. What is the first step in differentiating y = 8 ln x - 9 x with respect to x^2?

The first step is to rewrite the equation using the power rule, which states that the derivative of x^n is equal to n*x^(n-1).

2. How do I differentiate the natural logarithm term in the equation?

The derivative of ln x is equal to 1/x. Therefore, the natural logarithm term in the equation can be rewritten as 8/x.

3. What should I do with the constant term (-9) when differentiating?

The constant term does not affect the derivative, so it can be ignored when differentiating.

4. How do I differentiate the x term in the equation?

The derivative of x is equal to 1. Therefore, the x term in the equation can be rewritten as 1*x.

5. What is the final derivative of y = 8 ln x - 9 x with respect to x^2?

The final derivative is 8/x - 18/x^2.

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