find the derivative of the function ( f(y)=cosh(cosh(6y)) ) ( f(y)= )

Explanation:

Step1: Apply the chain rule

Let (u = \cosh(6y)), then (f(y)=\cosh(u)). The chain rule states (\frac{df}{dy}=\frac{df}{du}\cdot\frac{du}{dy}). First, (\frac{df}{du}=\sinh(u)) (since the derivative of (\cosh(x)) is (\sinh(x))).

Step2: Find (\frac{du}{dy})

Now, for (u = \cosh(6y)), let (v = 6y). Then (u=\cosh(v)). Using the chain rule again, (\frac{du}{dy}=\frac{du}{dv}\cdot\frac{dv}{dy}). We know (\frac{du}{dv}=\sinh(v)) and (\frac{dv}{dy} = 6). So (\frac{du}{dy}=6\sinh(6y)).

Step3: Substitute back

Substitute (u=\cosh(6y)) into (\frac{df}{du}\cdot\frac{du}{dy}). We get (f^{\prime}(y)=\sinh(\cosh(6y))\cdot6\sinh(6y)).

Answer:

(6\sinh(6y)\sinh(\cosh(6y)))