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Question

Find the derivative of the following functions from first principle:
cos(xπ8)

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Solution

Let f(x)=cos(xπ8)
Thus using first principle
f(x)=limx0f(x+h)f(x)h
=limx01h[cos(x+hπ8)cos(xπ8)]
=limx01h⎢ ⎢2sin(x+hπ8+xπ8)2sin(x+hπ8x+π82)⎥ ⎥
=limx01h[2sin(2x+hπ42)sinh2]
=limx0⎢ ⎢sin(2x+hπ42)sin(h2)(h2)⎥ ⎥
=sin(2x+0π42)1=sin(xπ8)

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