First we know that,

A=w*l

Now A,w,l are function of t (time)

Thus, (differenciating and using product rule),

dA/dt=w*dl/dt+l*dw/dt

We are given that,

dl/dt=-2 cm/sec

dw/dt=2 cm/sec

And, w=5 cm with l=12 cm

Thus,

dA/dt=(5)(-2)+12(2)=10 cm^2/sec

For perimeter consider,

P=2w+2l

Thus,

dP/dl=2(dw/dt)+2(dl/dt)

Thus,

dP/dl=2(2)+2(-2)=0 cm/sec

Which makes sense since if you are increasing one and decreasing one at the same rate there is no change in rate.

For the diagnol, (by Pythagorean theorem)

d^2=l^2+w^2

Thus, (implicit differenciation)

2d*(dd/dt)=2l(dl/dt)+2w(dw/dt)

When (l,w)=(5,12) we have d=13

2(13)(dd/dt)=2(5)(-2)+2(12)(2)

Thus,

26(dd/dt)=28

Thus,

dd/dt=28/26 cm/sec