Derivative of y=x*sqrt(x)*(4ln(5x)-6)

1. Apply the product rule:

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

   $f{\left(x \right)} = \sqrt{x} x$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the product rule:

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

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

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

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

      1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

      The result is: $\frac{3 \sqrt{x}}{2}$

   $g{\left(x \right)} = 4 \log{\left(5 x \right)} - 6$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $4 \log{\left(5 x \right)} - 6$ term by term:

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

         1. Let $u = 5 x$.

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

         3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$:

            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: $5$

            The result of the chain rule is:

            $$\frac{1}{x}$$

         So, the result is: $\frac{4}{x}$

      2. The derivative of the constant $-6$ is zero.

      The result is: $\frac{4}{x}$

   The result is: $\frac{3 \sqrt{x} \left(4 \log{\left(5 x \right)} - 6\right)}{2} + 4 \sqrt{x}$

2. Now simplify:

   $$\sqrt{x} \left(6 \log{\left(5 x \right)} - 5\right)$$

The answer is:

$$\sqrt{x} \left(6 \log{\left(5 x \right)} - 5\right)$$