Find the derivative of tan-1cosx1+sinx with respect to sec-1(x).
Find the required derivative.
Given function : tan-1cosx1+sinx
Let u=tan-1cosx1+sinx
z=cosx1+sinxz=cos2x2-sin2x2sin2x2+cos2x2+2Ă—sinx2.cosx2∵cosθ=cos2θ2-sin2θ2,sinθ=2Ă—sinθ2.cosθ2andsin2θ2+cos2θ2=1z=cosx2+sinx2cosx2-sinx2cosx2+sinx22z=cosx2-sinx2cosx2+sinx2z=1-tanx21+tanx2z=tanÏ€4-x2
Let v=sec-1x
differentiating v with respect to x we get,
dudv=1xx2-1
dudx=-12
Therefore,dvdu=dvdx.dxdu
=-12xx2-1
Hence, the derivative of tan-1cosx1+sinx with respect to sec-1(x) is -12xx2-1.