Derivative of y=log^2x+4x

1. Differentiate $4 x + \log{\left(x \right)}^{2}$ term by term:

   1. Let $u = \log{\left(x \right)}$.

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

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

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

      The result of the chain rule is:

      $$\frac{2 \log{\left(x \right)}}{x}$$

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

      1. Apply the power rule: $x$ goes to $1$

      So, the result is: $4$

   The result is: $4 + \frac{2 \log{\left(x \right)}}{x}$

The answer is:

$$4 + \frac{2 \log{\left(x \right)}}{x}$$