Differentiate
\[
y=3^{x^{4}+3}
\]

Step 1 ：Given the function \(y=3^{x^{4}+3}\).

Step 2 ：We need to find its derivative.

Step 3 ：The given function is a composition of several functions, so we will need to use the chain rule to differentiate it.

Step 4 ：The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 5 ：In this case, the outer function is \(3^u\) and the inner function is \(x^4 + 3\).

Step 6 ：The derivative of \(3^u\) with respect to \(u\) is \(3^u \ln(3)\), and the derivative of \(x^4 + 3\) with respect to \(x\) is \(4x^3\).

Step 7 ：So, the derivative of the given function is \(3^{x^4 + 3} \ln(3) \cdot 4x^3\).

Step 8 ：Final Answer: The derivative of the function \(y=3^{x^{4}+3}\) is \(\boxed{4x^{3}3^{x^{4}+3}\ln(3)}\)