Differentiate the following functions `w.r.t.x` from the first principle : `x^2 cos x`

Explanation

Let `y =x^(2) cos x`.
Let `deltay` be an increment in y, corresponding to an increment to an increment `deltax` in x.
Then `y +deltay = (x+deltax)^(2)+ cos (x + deltax)`
`rArr deltay = (x+deltax)^(2) cos (x+deltax) - x^(2) cos x`
`rArr (deltay)/(deltax) = ((x+deltax)^(2) cos (x+deltax)-x^(2) cosx)/(deltax)`
`rArr (dy)/(dx) = underset(deltaxrarr 0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr 0)("lim") ((x+deltax)^(2) cos (x+deltax) - x^(2) cos x)/(deltax)`
`= underset(deltaxrarr0)("lim") ((x^(2)+deltax^(2) + 2xdeltax)cos(x+deltax)-x^(2)cosx)/(deltax)`
`=(x^(2)[cos(x+deltax)-cosx]+(deltax)^(2)cos(x+deltax)+2x.deltax.cos(x+deltax))/(deltax)`
`= underset(deltaxrarr0)("lim"){(x^(2)[-2sin(x+(deltax)/(2))sin'(deltax)/(2)]+(deltax)^(2)cos(x+deltax)+2x.deltax.cos(x+deltax))/(deltax)}`
`= underset(deltax rarr 0) ("lim") [-x^(2)sin(x+(deltax)/(2)).(sin((deltax)/(2)))/(((deltax)/(2)))+(deltax) cos(x+deltax) +2xcos (x+deltax)]`
`= -x^(2).underset(deltaxrarr0)("lim")sin (x+(deltax)/(2)).underset(deltaxrarr0)("lim")(sin((deltax)/(2)))/(((deltax)/(2))) + underset(deltaxrarr0)("lim") (deltax) cos (x+deltax) + 2x.underset(deltax rarr0)("lim") cos (x+deltax)`
`= (-x^(2) xx sin x xx 1) + 0 + 2x cos x = (-x^(2) sinx + 2x cosx)`.
Hence, `(d)/(dx) (x^(2)cos x) = (-x^(2) sinx + 2 x cosx)`.