Differentiate each of the following from first principle: `xsinx`

Explanation

Let `y = x sin x`.
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then, `y + deltay = (x + deltax) sin (x + deltax)`
`rArr (deltay)/(deltax) = ((x+deltax) sin (x+deltax) - xsinx)/(deltax)`
`rArr (deltay)/(deltax) =underset(deltax rarr0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr0)("lim") ([(x+deltax) sin (x+delta) - xsinx])/(deltax)`
`= underset(deltax rarr0)("lim') [(x[sin(x+deltax) - sinx])/(deltax) + sin (x+delta)]`
`= underset(deltaxrarr0)("lim") (x[sin (x+deltax) =- sin x])/(deltax) + underset(deltax rarr0)("lim") sin (x+delta)`
`=underset(deltaxrarr0)("lim"){(2x(x+(deltax)/(2))sin((deltax)/(2)))/(deltax)}+sinx`
`= {x . underset(deltaxrarr0)("lim")cos(x+deltax)/(2).underset(deltararrx)("lim') (sin (deltax//2))/((delta//2))} + sin x`
`= (x.cosx.1) + sin x = (x cos x + sinx)`.
Hence, `d/(dx)(xsinx) = (xcosx + sinx)`.