a Use the Quotient Rule to differentiate the function fxtanx

(a) Use the Quotient Rule to differentiate the function f(x)=tan(x)-1/sec(x). f\'(x)=

(b) Simplify the expression for f(x) by writing it in terms of sin(x) and cos(x), the find f\'(x).

(c) Are your answers to part (a) and (b) equivalent?

For part (a) (which I got correct), I got f\'(x)=-sin(x)tan(x)+sec(x)+sin(x)

I got part (b) wrong and I am certain it is a simple thing I am overlooking, but if someone could show me, I would appreciate it.

For part (c) I knew they were equivalent, and the answers is yes.

Solution

a) (tanx - 1)/secx f\'(x) = (sec x (sec^2 x) - (tan x - 1) sec x tan x)/(sec^2 x) = (sec^3 x - sec x(sec^2 x - 1) + sec x tan x)/sec^2 x = (1+tanx)/secx = cosx + sinx b)In terms of sin x and cos x f(x) = (tanx - 1)/secx we get sin x - cos x differentiating we get cos x + sin x so both the answers are same