Question

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:

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