what is the 2021st derivative of f(x)=x^(2020)+e^x+sin(x)?

Answer 1

Suppose you have f(x) = x^3
f' = 3x^2
f" = 3*2*x^1
f"' = 3*2*1 = 3!
That should give you a clue for the 1st term

e^x stays the same

sin(x) goes through a 4-step cycle: cos, -sin, -cos, sin
since 2021 = 4*505+1, the 2021st derivative will be cos(x)

Answer 2

so it is it 2020!x^n for the first term?

Answer 3

Not quite
No, the 2020th derivative is 2020!, which is a constant!
so, the nth derivative of x^m is zero if n > m