Derivative of y=(lnx+2x)(3x^3-17x)

1. Apply the product rule:

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

   $f{\left(x \right)} = 2 x + \log{\left(x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

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

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

      2. 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: $2$

      The result is: $2 + \frac{1}{x}$

   $g{\left(x \right)} = 3 x^{3} - 17 x$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $3 x^{3} - 17 x$ 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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $9 x^{2}$

      2. 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: $-17$

      The result is: $9 x^{2} - 17$

   The result is: $\left(2 + \frac{1}{x}\right) \left(3 x^{3} - 17 x\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$

2. Now simplify:

   $$\left(2 x + 1\right) \left(3 x^{2} - 17\right) + \left(2 x + \log{\left(x \right)}\right) \left(9 x^{2} - 17\right)$$

The answer is: