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The chain rule needs to be applied when you are differentiating a function f of x with respect to a variable t, x is a function of (i.e. given f(x) and x(t), you want to calculate df/dt. 

To do so ...

*Use identity sinxcosx = 1/2sin2x

=5/2sin2x + 5cosx

*then use simple integration gives

=(-5/2 x1/2)cosx + 5sinx + c

= =5/4cosx +5sinx + c

5 + 5√ 2 - √ 8 - √ 16    (Break up all terms in terms of square roots of 2)

5 + 5√ 2 - √ 8 - 4

5 + 5√ 2 - 2 √ 2 - 4 

1 + 3√2

For this question, one would sum the algebraic lengths of the rectangle, creating a result of 4x + 16.

This is the algebraic perimeter of the rectangle which must be equated to the numerical value ...

Sinx differentiates to give cosx.

We then multiply this by the differential of "2x."

Which equals 2.

Our answer therefore is "2cos(2x)."