Need help finding derivatives?

Use Product Rule:

fx = (fg) (fh)

f' x = [(fg) (f' h)] + [(fh) (f' g)]

f' x = [(4x + 5e^x) (1 - e^x)] + [(x - e^x) (4+5e^x)]

Expand:

f' x = [4x + 5e^x + 4xe^x - 5e^2x] + [4x + 5xe^x - 4e^x - 5e^2x]

Simplify:

f' x = [8x - 10e^2x + 9xe^x + e^x]

Final answer:

f' x = 8x - 10e^2x + 9xe^x + e^x

Hope that helps.

f(x) = [4x + 5.e^(x)].[x - e^(x)]

This function looks like (u.v), so its derivative looks like: (u'.v) + (v'u) → where:

u = 4x + 5.e^(x) → u' = 4 + 5.e^(x)

v = x - e^(x) → v' = 1 - e^(x)


f'(x) = (u'.v) + (v'u)

f'(x) = [4 + 5.e^(x)].[x - e^(x)] + [1 - e^(x)].[4x + 5.e^(x)]

f'(x) = 4x - 4.e^(x) + 5x.e^(x) - 5.e^(x).e^(x) + 4x + 5.e^(x) - 4x.e^(x) - 5.e^(x).e^(x)

f'(x) = 8x + x.e^(x) + e^(x) - 10.e^(x).e^(x)

f'(x) = 8x + x.e^(x) + e^(x) - 10.e^(2x)

f'(x) = 8x - 10.e^(2x) + (x + 1).e^(x)