#dy/dx(-3e^-xsec(x))#
Let's factor #-3# outside of the derivative. This cleans up the calculations later on, and we can do it because it's just a constant.
#-3d/dxe^-xsec(x)#
Recall that the derivative of a function in the form #(fg), (fg)',# is given by #fg'+gf'#.
Here, we can say
#f(x)=e^-x#
#g(x)=sec(x)#
#f'(x)=e^-x(d/dx(-x))=-e^-x# (Chain Rule -- #d/dxe^(h(x))=e^(h(x))d/dx(h(x))# where #h(x)# is a function.
#g'(x)=sec(x)tan(x)# (This is a common derivative; memorize it. It can be derived from #sec(x)=1/cos(x)# using the Quotient Rule as well)
y'=#fg'+gf'=-3(e^-xsec(x)tan(x)+(-sec(x)e^-x)=e^-xsec(x)tan(x)-e^-xsec(x))#
Do not forget to multiply everything by #-3,# the constant we factored out.
We can simplify a bit by factoring out #e^-xsec(x):#
#y'=-3e^-xsec(x)(tan(x)-1)#
