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Question

Differentiate secx by first principle.

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Solution

Let f(x)=secx

f(x+h)sec(x+h)

ddxf(x)=limh0[f(x+h)f(x)h]

ddx(secx)=limh0[sec(x+h)secxh]

ddx(secx)=limh0[1h(1cos(x+h)1cosx)]

ddx(secx)=limh0[1h(cos(x)cos(x+h)cos(x+h)cosx)]

ddx(secx)=limh0⎢ ⎢ ⎢ ⎢ ⎢ ⎢1h2sin(2x+h2)sin(x+hx2)h.cos(x+h)cosx⎥ ⎥ ⎥ ⎥ ⎥ ⎥

and ddx(secx)=limh0⎢ ⎢ ⎢ ⎢ ⎢ ⎢sin(x+h2)cos(x+h)cosx⎥ ⎥ ⎥ ⎥ ⎥ ⎥ limh0⎢ ⎢ ⎢ ⎢ ⎢ ⎢sinh2h2⎥ ⎥ ⎥ ⎥ ⎥ ⎥

ddx(secx)=sinxcosxcosx×1

ddx(secx)=secxtanx.


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