Use part I of the Fundamental Theorem of Calculus, to find the derivative of \[ \begin{array}{l} h(x)=\int_{-5}^{\sin (x)}\left(\cos \left(t^{4}\right)+t\right) d t \\ h^{\prime}(x)= \end{array} \]

Step 1 ：Given the function \(h(x)=\int_{-5}^{\sin (x)}\left(\cos \left(t^{4}\right)+t\right) d t\), we are asked to find its derivative \(h'(x)\).

Step 2 ：We use the Fundamental Theorem of Calculus Part I, which states that if a function f is continuous over the interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a).

Step 3 ：In this case, we are asked to find the derivative of a function defined by an integral, which is a direct application of the Fundamental Theorem of Calculus Part I.

Step 4 ：The derivative of the integral from a constant to a function of x is simply the integrand evaluated at that function of x, multiplied by the derivative of that function of x.

Step 5 ：In this case, the function of x is \(\sin(x)\), and the integrand is \(\cos(t^{4}) + t\). So, we need to substitute \(\sin(x)\) for t in the integrand, and then multiply by the derivative of \(\sin(x)\), which is \(\cos(x)\).

Step 6 ：Substituting \(\sin(x)\) for t in the integrand, we get \(\sin(x) + \cos(\sin(x)^{4})\).

Step 7 ：Multiplying this by the derivative of \(\sin(x)\), which is \(\cos(x)\), we get \((\sin(x) + \cos(\sin(x)^{4}))\cos(x)\).

Step 8 ：So, the derivative of the function \(h(x)\) is \(\boxed{(\sin(x) + \cos(\sin(x)^{4}))\cos(x)}\).