If y=\arctan(\frac{2x−1}{1+x−x^2}), then \frac{dy}{dx} at x=1 is equal to? I have

stropa0u 2022-01-24 Answered
If y=arctan(2x11+xx2), then dydx at x=1 is equal to?
I have tried 2 different approaches, both yielding different results and both results are present in the options. Am I doing something wrong or why is this happening?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

bekiffen32
Answered 2022-01-25 Author has 8 answers
In the derivative of
y=arctan(x)arctan(1x)+kπ
dydx=11+x2(11+(1x)2)
which is the same as in the second approach.
Also note that the inverse tangent is an odd function, so that arctan(1x)=arctan(x1), so that there is no structural difference between the two approaches
Not exactly what you’re looking for?
Ask My Question
Fallbasisz8
Answered 2022-01-26 Author has 9 answers
A computationally simpler method of directly calculating the derivative at x=1 is to write
tany=2x11+xx2
so that
logtany=log(2x1)log(1+xx2)
Then implicit differentiation yields
sec2ytanydydx=22x112x1+xx2
Since, when x=1, we have tany=211+11=1, it follows from the trigonometric identity sec2y=1+tan2y  that  sec2y=1+12=2; therefore
2[dydx]x=1=221121+11=3
hence the answer is 3/2. Note this solution uses two tactics; logarithmic differentiation and implicit differentiation, to make the calculation extremely simple.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more