Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. \[ g(x)=\int_{6}^{x} t^{5} \sin (t) d t \]

Step 1 ：Given the function \(g(x)=\int_{6}^{x} t^{5} \sin (t) d t\), we are asked to find its derivative.

Step 2 ：According to the Fundamental Theorem of Calculus Part I, the derivative of a function defined by an integral from a to x of f(t) dt is simply f(x).

Step 3 ：Therefore, the derivative of the given function is simply the integrand evaluated at x, which is \(x^{5} \sin (x)\).

Step 4 ：\(\boxed{x^{5} \sin (x)}\) is the final answer.