Find the derivative

Question

f(x)=ln(x/x²+1)

I know we use the chain rule, but I'm stuck right after that.

Answer 1

Instead of using the Chain Rule right away, we can first simplify the function by using log properties.

f(x) = ln(x / (x²+1)) = lnx - ln(x² + 1)

So, f'(x) = 1/x - 2x / (x²+1) = (x²+1-2x) / [x(x²+1)] =

(x-1)² / [x(x²+1)]

Answer 2

To use the chain rule, we take the derivative of the fraction inside the natural log, then multiply it with the outer derivative. We have:

(d/dx)(x/(x² + 1)) = (x² + 1 - 2x²)/(x² + 1)² = (1-x²)/(x² + 1)² (Quotient Rule)

The outer derivative is 1/(x/(x² + 1)) = (x² + 1)/x, therefore the final answer is:

f'(x) = (1-x²)(x² + 1)/[(x² + 1)²x] = (1 - x²)/[x(x² + 1)]