How do you differentiate #f(x) = 4/sqrt(tan^2(1-x) # using the chain rule?
See the explanation section below.
To avoid the quotient rule, let's rewrite
Now use the power and chain rules:
# = 4[(-1/2)(tan^2(1-x))^(-3/2)][2(tan(1-x))][sec^2(1-x)]d/dx(1-x)#
# = 4[(-1/2)(tan^2(1-x))^(-3/2)][2(tan(1-x))][sec^2(1-x)][-1]#
Simplifying algebraically, gets us,
# = (4tan(1-x)sec^2(1-x))/(tan^2(1-x))^(3/2)#
but if we square first, we get
If you don't mind rewriting as a piecewise function, you could use:
Now differentiate each piece to get a piecewise derivative.