Derivative of x^4(12lnx-7)

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{4}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$

   $g{\left(x \right)} = 12 \log{\left(x \right)} - 7$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $12 \log{\left(x \right)} - 7$ term by term:

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

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

         So, the result is: $\frac{12}{x}$

      2. The derivative of the constant $-7$ is zero.

      The result is: $\frac{12}{x}$

   The result is: $4 x^{3} \left(12 \log{\left(x \right)} - 7\right) + 12 x^{3}$

2. Now simplify:

   $$x^{3} \left(48 \log{\left(x \right)} - 16\right)$$

The answer is:

$$x^{3} \left(48 \log{\left(x \right)} - 16\right)$$