Question

find the derivative of ( f(x) ).

( f(x)=\tan (-4 x-8) )

( 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:

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