Derivative of x^4lnx

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 $4 x^{3}$

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

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

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

2. Now simplify:

   $$x^{3} \cdot \left(4 \log{\left(x \right)} + 1\right)$$

The answer is:

$$x^{3} \cdot \left(4 \log{\left(x \right)} + 1\right)$$