Derivative of ln5*log5(x^2+1)

1. Apply the quotient rule, which is:

   $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

   $f{\left(x \right)} = \log{\left(5 \right)} \log{\left(x^{2} + 1 \right)}$ and $g{\left(x \right)} = \log{\left(5 \right)}$.

   To find $\frac{d}{d x} f{\left(x \right)}$:

   1. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let $u = x^{2} + 1$.

      2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$.

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$:

         1. Differentiate $x^{2} + 1$ term by term:

            1. Apply the power rule: $x^{2}$ goes to $2 x$

            2. The derivative of the constant $1$ is zero.

            The result is: $2 x$

         The result of the chain rule is:

         $$\frac{2 x}{x^{2} + 1}$$

      So, the result is: $\frac{2 x \log{\left(5 \right)}}{x^{2} + 1}$

   To find $\frac{d}{d x} g{\left(x \right)}$:

   1. The derivative of the constant $\log{\left(5 \right)}$ is zero.

   Now plug in to the quotient rule:

   $\frac{2 x}{x^{2} + 1}$

2. Now simplify:

   $$\frac{2 x}{x^{2} + 1}$$

The answer is:

$$\frac{2 x}{x^{2} + 1}$$