Derivative of 4x^2log0,5x

1. Apply the product rule:

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

   $f{\left(x \right)} = 4 x^{2}$; 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. Apply the power rule: $x^{2}$ goes to $2 x$

      So, the result is: $8 x$

   $g{\left(x \right)} = \log{\left(0.5 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = 0.5 x$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 0.5 x$:

      1. 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: $0.5$

      The result of the chain rule is:

      $$\frac{1.0}{x}$$

   The result is: $8 x \log{\left(0.5 x \right)} + 4.0 x$

2. Now simplify:

   $$x \left(8.0 \log{\left(x \right)} - 1.54517744447956\right)$$

The answer is:

$$x \left(8.0 \log{\left(x \right)} - 1.54517744447956\right)$$