Find the derivative of f(x)=ln(x)/x^2+1. Do not simplify. 

We are asked to find the derivative of the function:

\[
f(x) = \frac{\ln(x)}{x^2 + 1}
\]

To solve this, we will use the quotient rule. The quotient rule states that if we have a function in the form:

\[
\frac{u(x)}{v(x)}
\]

The derivative is given by:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}
\]

Here, we have:

\[
u(x) = \ln(x), \quad v(x) = x^2 + 1
\]

Now, we need to find \(u'(x)\) and \(v'(x)\):

1. \( u'(x) = \frac{d}{dx} \ln(x) = \frac{1}{x} \)
2. \( v'(x) = \frac{d}{dx} (x^2 + 1) = 2x \)

Now apply the quotient rule:

\[
f'(x) = \frac{ \left( \frac{1}{x} \right)(x^2 + 1) - (\ln(x))(2x)}{(x^2 + 1)^2}
\]

Thus, the derivative is:

\[
f'(x) = \frac{ \frac{x^2 + 1}{x} - 2x \ln(x)}{(x^2 + 1)^2}
\]

This is the derivative without any simplification.

Would you like more details on any step or have other questions?

Here are 5 related questions to extend your understanding:
1. How would you simplify the derivative we just found?
2. What are the critical points of the function \( f(x) = \frac{\ln(x)}{x^2 + 1} \)?
3. How would you find the second derivative of \( f(x) \)?
4. How does the derivative behave as \( x \to \infty \)?
5. How do the quotient rule and product rule differ?

**Tip:** The quotient rule can be easier if you recognize when you can simplify the expressions beforehand.

Find the derivative of f(x)=ln(x)/x^2+1. Do not simplify.

This page explains how to find the derivative of f(x) = ln(x)/(x^2 + 1) using the quotient rule. Follow a step-by-step solution without simplification, covering logarithmic functions and differentiation techniques. Ideal for students in advanced calculus (Grades 11-12).

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