Use the chain rule to find the derivative of $f(x)=2 e^{-4 x^{10}-2 x^{3}}$

Step 1 ：Let's find the derivative of the function \(f(x)=2 e^{-4 x^{10}-2 x^{3}}\).

Step 2 ：We use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

Step 3 ：In this case, the outer function is \(2e^x\) and the inner function is \(-4x^{10}-2x^{3}\).

Step 4 ：The derivative of the outer function \(2e^x\) is \(2e^x\).

Step 5 ：The derivative of the inner function \(-4x^{10}-2x^{3}\) is \(-40x^{9} - 6x^{2}\).

Step 6 ：Multiplying these two derivatives together, we get the derivative of the function \(f(x)=2 e^{-4 x^{10}-2 x^{3}}\) as \(2(-40x^{9} - 6x^{2})e^{-4x^{10} - 2x^{3}}\).

Step 7 ：Final Answer: \(\boxed{2(-40x^{9} - 6x^{2})e^{-4x^{10} - 2x^{3}}}\)