Question

differentiate ( f(x)=e^{cosh (8 x)} ).
( 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:

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