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Question
Find the derivative of the following function from first principle:
sin (x + 1)
Solution
Let f(x) = sin (x + 1)
We have, `d/(dx)` (f(x)) = `lim_(h->0) ([f(x + h) - f(x)])/h`
= `lim_(h->0) (sin (x + h + 1) - sin (x + 1))/h`
= `lim_(h->0)(2 cos ((x + h + 1 + x + 1)/2) sin ((x + h + 1 - x - 1)/2))/h`
= `lim_(h->0)(2[cos ((2(x + 1) + h)/2)sin h/2])/(2 xx (h/2))`
= `lim_(h->0) cos (x + 1 + h/2) ((sin h/2)/ (h/2))`
= `{lim_(h->0) (x + 1 + h/2)} {lim_(h->0) (sin h/2) / (h/2)}`
= cos (x + 1) × (1) = cos (x + 1)
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