Differentiate the following w.r.t. x.
a. `(x^2+3x)(2x+7)`
b. `(3x^2+2)(4x-3x^3)`
c. `sqrtx(x^3+x^2-3x)`
d. `sinxlogx`

Explanation

a. Let `y=(x^2+3x)(2x+7)`
`(dy)/(dx)=(x^2+3x)(d(2x+7))/(dx)+[(d)/(dx)(x^2+3x)](2x+7)`
`=(x^2+3x)2+(2x+3)(2x+7)=6x^2+26x+21`
b. Let `y=(3x^2+2)(4x-3x^3)`
`(dy)/(dx)=(3x^2+2)(d)/(dx)(4x-3x^3)+[(d)/(dx)(3x^2+2)](4x-3x^3)`
`=(3x^2+2)(4-9x^2)+6x(4x-3x^3)=-45x^4+18x^2+8`
c. Let `y=sqrtx(x^3+x^2-3x)=x^(7//2)+x^(5//2)-3x^(3//2)`
`(dy)/(dx)=7/2x^(5//2)+5/2x^(3//2)-3xx3/2x^(1//2)`
`=(1)/(2sqrtx)(7x^3+5x^2-9x)`
d. Let `y=sinxlogx`
`(dy)/(dx)=sinx(d)/(dx)(logx)+[(d)/(dx)(sinx)]logx`
`=(sinx)/(x)+cosxlogx`