Question

How do you find the derivative of the function `y = sin(cos(sinx))`?

Answer 1

Use the chain rule repeatedly

Explanation:

Use the chain rule.

Chain rule 1:

`d/dx[sincossinx]=coscossinx*d/dx[cossinx]`

Chain rule 2:

`d/dx[cossinx]=-sinsinx*d/dx[sinx]=-sinsinx*cosx`

Combine:

`d/dx[sincossinx]=coscossinx*(-sinsinx)*cosx`

So

`dy/dx=-cosxsinsinxcoscossinx`

Answer 2

`y'=-cos(cos(sin(x))*sin(sin(x))*cos(x)`

Explanation:

We must use the chain rule,
so we get

`cos(cos(sin(x)))*(-sin(sin(x))*cos(x)`

Answer 3

`d/dxsin(cos(sinx))=-cos(cos(sinx))sin(sinx)cosx`

Explanation:

To find derivatives of composite functions, we multiply the derivatives of each function with respect to its arguments.

So

`d/dxsin(cos(sinx))=(dsin(cos(sinx)))/(dcos(sinx)) *(d(cos(sinx)))/(dsinx) * (dsinx)/dx`

`=cos(cos(sinx))(-sin(sinx))cosx`