How do you differentiate #f(x)= (1 - sin^2x)/(1 - sinx)^2 # using the quotient rule?

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Answer 1 · 2018-04-23T04:35:34

#(2cosx)/(1-sinx)^2#

First, factor the numerator.

#(1-sinx)(1+sinx)#

Then cancel a factor from the bottom.

#((1-sinx)(1+sinx))/((1-sinx)(1-sinx))#
#(1+sinx)/(1-sinx)#

We know the quotient rule:
#(f'(x)g(x)-g'(x)f(x))/g(x)^2#

So let's insert:
#(cosx*(1-sinx)-(-cosx)*(1+sinx))/((1-sinx)^2)#

We get in the numerator:
#(cosx*(1-sinx)+cosx*(1+sinx))#

#(cosx(1-sinx+1+sinx))#

#(cosx(2))#

Now in total:

#(2cosx)/(1-sinx)^2#

And it can be left this way. There is no more useful algebra to do. I double checked this answer.