differentiate ( f(x)=e^{cosh (8 x)} ).\n( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = e^{u}) and (u=\cosh(v)) and (v = 8x), then (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dv}\cdot\frac{dv}{dx}). First, (\frac{d}{du}(e^{u})=e^{u}).

Step2: Differentiate (\cosh(v))

The derivative of (\cosh(v)) with respect to (v) is (\sinh(v)).

Step3: Differentiate (v = 8x)

The derivative of (v = 8x) with respect to (x) is (8).

Step4: Substitute back

Substitute (u=\cosh(8x)) and (v = 8x) into the chain - rule formula. (\frac{df(x)}{dx}=e^{\cosh(8x)}\cdot\sinh(8x)\cdot8)

Answer:

(8e^{\cosh(8x)}\sinh(8x))