Question

the answer is given, I am interested in understanding the step-by-step details leading up to the answer.

Answer 1

Given: The function below

`y=(e^{cos((t)/(9))})^4`

To Determine: The derivative of the given function

Solution

Let us apply chain rule

`(dy)/(dx)=(dy)/(du)*(du)/(dx)`

The equation becomes

`y=e^{4cos((t)/(9))}`
`\begin{gathered} u=4cos((t)/(9)) \\ x=(t)/(9) \\ (dx)/(dt)=(1)/(9) \\ u=4cosx \\ (du)/(dx)=-4sinx \\ (du)/(dt)=(du)/(dx)*(dx)/(dt) \\ (du)/(dt)=-4sinx*(1)/(9) \\ (du)/(dt)=-(4)/(9)sin((1)/(9)) \end{gathered}`
`\begin{gathered} y=e^u \\ (dy)/(du)=e^u \\ (dy)/(dt)=(dy)/(du)*(du)/(dt) \\ (dy)/(dt)=e^u*-(4)/(9)sin((1)/(9)) \\ (dy)/(dt)=e^{4cos((t)/(9))}*-(4)/(9)sin((1)/(9)) \\ (dy)/(dt)=-(4)/(9)sin((t)/(9))e^{cos((t)/(9))} \end{gathered}`

Hence,

`(dy)/(dt)=-(4)/(9)sin((t)/(9))e^{cos((t)/(9))}`