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))