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Question

Find the derivative of tan-1cosx1+sinx with respect to sec-1(x).


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Solution

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

differentiating v with respect to x we get,

dudx=-12

Therefore,dvdu=dvdx.dxdu

=-12xx2-1

Hence, the derivative of tan-1cosx1+sinx with respect to sec-1(x) is -12xx2-1.


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