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Question

Use the quotient rule, or otherwise, to prove that
ddx(cosechx)=cosechxcothx

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Solution

We know that cosechx=1sinhx
Hence ddxcosechx=ddx1sinhx
Differentiate the given equation with respect to x using quotient rule
ddx1sinhx=[sinhx(d/dx(1))(1)(d/dx(sinhx)]sinh2x=0coshxsinh2x=coshxsinh2x=coshxsinhxsinhx=coshxsinhx1sinhx=cothxcosechx
Hence proved that ddxcosechx=cothxcosechx

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