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# y= tan^-1(secx-tanx), then dy/dx AskIITian Expert Priyasheel - IITD
8 Points
11 years ago

d (tan-1(x))/dx = 1/(1+x^2)
To prove the above result,
let y=tan-1(x)
or,tan(y)=x, differentiate w.r.t. x,
sec^2(y).dy/dx=1, or,
dy/dx=cos^2(y) ...(i)
Let p=tan^-1(x), or x=tan(p), so cos(p)=1/(1+x^2)^0.5,
So, cos(p)=cos(tan-1(x))=1/(1+x^2)^0.5.
Therefore, dy/dx=cos^2(y)=cos^2(tan-1(x))=1/(1+x^2)^0.5

In the original question
dy/dx=(sec(x)tan(x)-sec^2(x)) / (1 + (sec(x) - tan(x))^2)
=-1/2 (after simplification, using 1+tan^2(x)=sec^2(x))

11 years ago

derivative of arc tan x is 1/(1+x2)

here dy/dx= ( secx.tanx - sec2x) / (1+sec2x+tan2x-2.secx.tanx)

= secx(secx-tanx)/2.tanx(secx-tanx)

=secx/2tanx

=(cosec x) /2

--

regards

Ramesh