Differentiate each of the functions with respect to ‘x’

Differentiate using first principle x cos x.

Let y = x Cosx ----- (i)

y + ∆y = (x + ∆x) Cos (x + ∆x) ----- (ii)

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

y + ∆y – y = (x + ∆x) Cos (x + ∆x) - x Cosx

∆y = xCos (x + ∆x) + ∆x Cos (x + ∆x) – x Cosx

Dividing both sides by ∆x and take the limits,

Taking the limits,

=x(-Sin x)+Cos x

= -x Sinx + Cos x

Hence the answer is -x Sinx + Cos x.