F.5 Differentiation

Let tanA = 2tanX and y = tan(A-X) where 0<=X<兀/2
(a) Express y in terms of tanX.
(b) When dy/dx =0 , find the value of x.
(a)
y
= tan(A-X)
=(tanA-tanX)/(1+tanAtanX)
=tanX/(1+2tan^2X)
(b)
dy/dx =0
d/dx [tanX/(1+2tan^2X)] =0
d/du [u/1+2u^2](du/dx)=0 [where u=tanX]
{[(1+2u^2)-4u^2]/(1+2u^2)^2}(sec^2x)=0
(1-2u^2)/(1+2u^2)^2=0
1-2u^2=0
u^2=1/2
u=1/√2
tanX=1/√2
X=35.264 degrees