Differentiate the following functions with respect to `x`
(a) `x^(3) + 5x^(2) - 2`
(b) `x sin x`
(c) `(2x + 3)^(6)`
(d) `(x)/ (sinx)`
(e) `e^((5x+2))`

Explanation

(a) `(d)/(dx)(x^(3)+5x^(2)-2) = (d)/(dx) (x^3) + 5 (d)/(dx) (x^(2)) - (d)/(dx) (2)`
`= 3x^(2) + 5(2x) - 0`
`= 3x^(2)+ 10x`
(b) `(d)/(dx) (xsin x) =x(d)/(dx)(sin x) + sin x.(d)/(dx) (x)`
`= x cos x + sin x (1)`
`= x cos x + sin x`
(C) `(d)/(dx) (2x+3)^(6) =2 (d)/(dX) (X)^(6 )`, where `X = 2x + 3`
= `2{6X^5} = 12X^(5) = 12 ( 2x+3)^(5)`
(d) `(d)/(dx) ((x)/(sinx)) =(sinx(d)/(dx)(x)-x(d)/(dx)(sinx))/(sinx)^(2)`
=`((sinx)(1)- x (cosx))/(sin^2x) = (sinx- x cos x)/ (sin^2x)`
(e) `(d)/(dx) e^((5x+2)) = 5(d)/(dx) e^(X)` , where `X =5x +2 = 5e^(X) =5e^(5x+2)`