find the derivative of ( f(x) ).\n( f(x)=sec (3 x-7) )\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 = 3x-7), so (f(x)=\sec(u)). The derivative of (\sec(u)) with respect to (u) is (\sec(u)\tan(u)).

Step2: Differentiate the inner function

Differentiate (u = 3x - 7) with respect to (x). Using the power rule ((ax + b)^\prime=a), we get (u^\prime=3).

Step3: Combine the results

By the chain rule (f^\prime(x)=\frac{d}{du}(\sec(u))\cdot\frac{du}{dx}). Substituting (u = 3x-7), (\frac{d}{du}(\sec(u))=\sec(u)\tan(u)) and (\frac{du}{dx}=3), we have (f^\prime(x)=3\sec(3x - 7)\tan(3x - 7)).

Answer:

(3\sec(3x - 7)\tan(3x - 7))