Derivative of x^4*4^x

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 $4 x^{3}$

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

   1. $\frac{d}{d x} 4^{x} = 4^{x} \log{\left(4 \right)}$

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

2. Now simplify:

   $$x^{3} \left(2^{2 x + 2} + \log{\left(4^{4^{x} x} \right)}\right)$$

The answer is:

$$x^{3} \left(2^{2 x + 2} + \log{\left(4^{4^{x} x} \right)}\right)$$