Find the derivatives of the functions.
(a) `f(x) =x^8`
(b) `f(x) =x^3 + sin x`
`f(x) =x^3 sin x`

Explanation

(a) Using `(dx^n)/(dx) =nx^(n-1)`
`(d(x^8))/(dx) =8x^7`
(b) Using `d/(dx) (f_1(x)+f_2(x))=(df_1(x))/(dx) +(df_2(x))/(dx) , (d(sin x))/(dx) =cos x`
`d/dx (x^3+sin x) = (d(x^3))/(dx) +(d(sinx))/(dx) =3x^2 + cos x`
(c ) Using `d/(dx)(f_1 (x) f_2 (x)) = f_1 (x) (df_2(x))/(dx) + (df_1(x))/(dx) f_2 (x) and (d(sinx))/(dx) =cos x`
`d/(dx) (x^3 sin x) =x^3 (d(sin x))/(dx) + (d(x^3))/(dx) sin x = x^3 cos x +3x^2 sin x`