Answer :

Let f(x) = Cos (x2 + 1) ------------(i)


f(x + ∆x) = Cos [(x + ∆x)2 + 1] -------(ii)


Subtracting eq. (i) from eq. (ii),


f(x + ∆x) - f(x) = Cos [(x + ∆x)2 + 1] - Cos (x2 + 1)


Dividing both sides by ∆x,





As per the definition of differentiations,












Taking limits,


= -2 Sin (x2 + 1).1.(x) = -2x Sin (x2 + 1)


As


Hence, the required answer is -2x Sin (x2 + 1).


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