find the derivative of ( f(x) ).\n\n( f(x)=\tan (-4 x-8) )\n\n( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). Let (u=-4x - 8), so (f(x)=\tan(u)). The derivative of (\tan(u)) with respect to (u) is (\sec^{2}(u)).

Step2: Find the derivative of the inner function

The derivative of (u=-4x - 8) with respect to (x) is (u^\prime=-4).

Step3: Combine the results

By the chain rule, (f^\prime(x)=\sec^{2}(-4x - 8)\cdot(-4))

Answer:

(-4\sec^{2}(-4x - 8))