Find the derivative of the following functions from first principle: sin(x+1)
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Solution
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Let f(x)=sin(x+1) Thus using first principle, f′(x)=limx→0f(x+h)−f(x)h =limx→01h[sin(x+h+1)−sin(x+1)] =limx→01h[2cos(x+h+1+x+12)sin(x+h+1−x−12)] =limx→01h[2cos(2x+h+22)sin(h2)] =limx→0⎡⎢
⎢⎣cos(2x+h+22)sin(h2)(h2)⎤⎥
⎥⎦ =cos(2x+0+22)⋅1=cos(x+1)
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