Answer 1

y = `x^4 + x sqrt(x) cos x - x^2"e"^x`

y = `x^4 + x^(3/2) cos x - x^2"e"^x`

Differentiating w.r.t. x, we get

`("d"y)/("d"x) = "d"/("d"x) (x^4 + x^(3/2) cos x - x^2 "e"^x)`

`("d"y)/("d"x) = "d"/("d"x) (x^4) + "d"/("d"x) (x^(3/2) cos x) - "d"/("d"x) (x^2"e"^x)`

= `4x^3 + x^(3/2) "d"/("d"x) cos x + cos x "d"/("d"x) (x^(3/2)) - [x^2 ("d"/("d"x) "e"^x) + "e"^x ("d"/("d"x) x^2)]`

= `4x^3 + x^(3/2) (- sin x) + cos x (3/2x^(1/2)) - [x^2 "e"^x + "e"^x (2x)]`

= `4x^3 - x^(3/2) sin x + 3/2 sqrt(x) cos x - x"e"^x (x + 2)`