Question

if (f(x)=5sec(7x)), find (f(x)). find (f(2)).

Explanation:

Step1: Recall the chain - rule

The chain - rule states that if \(y = f(g(x))\), then \(y'=f'(g(x))\cdot g'(x)\). Also, the derivative of \(\sec(u)\) with respect to \(u\) is \(\sec(u)\tan(u)\). Let \(u = 7x\), and \(y = 5\sec(u)\).

Step2: Differentiate the outer function

The derivative of \(y = 5\sec(u)\) with respect to \(u\) is \(y'_u=5\sec(u)\tan(u)\).

Step3: Differentiate the inner function

The derivative of \(u = 7x\) with respect to \(x\) is \(u'_x = 7\).

Step4: Apply the chain - rule

By the chain - rule \(f'(x)=\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). Substituting \(y'_u = 5\sec(u)\tan(u)\) and \(u'_x=7\) and \(u = 7x\) back in, we get \(f'(x)=5\sec(7x)\tan(7x)\cdot7 = 35\sec(7x)\tan(7x)\).

Step5: Find \(f'(2)\)

Substitute \(x = 2\) into \(f'(x)\): \(f'(2)=35\sec(14)\tan(14)\).

Answer:

\(f'(x)=35\sec(7x)\tan(7x)\)
\(f'(2)=35\sec(14)\tan(14)\)