Derivative of ln(5x²+2)

1. Let $u = 5 x^{2} + 2$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x^{2} + 2\right)$:

   1. Differentiate $5 x^{2} + 2$ term by term:

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

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

         So, the result is: $10 x$

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

      The result is: $10 x$

   The result of the chain rule is:

   $$\frac{10 x}{5 x^{2} + 2}$$

4. Now simplify:

   $$\frac{10 x}{5 x^{2} + 2}$$

The answer is:

$$\frac{10 x}{5 x^{2} + 2}$$