Differentiate each of the functions with respect to ‘x’

Differentiate using first principle cos (x² + 1)

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).