1. The problem is to differentiate the function $f(x) = 5x^6$ with respect to $x$. 2. Apply the power rule of differentiation, which states that if $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$. 3. Here, $a = 5$ and $n = 6$, so we compute: $$f'(x) = 5 \times 6 \times x^{6-1} = 30x^5$$ 4. Therefore, the derivative of $5x^6$ is $30x^5$.