Question

find the derivative of ( f(x) ).
( f(x)=sec (3 x-7) )
( 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:

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