Bonjour,
1)
f(x) = xeˣ + 3x - 1
u(x) = x
u'(x) = 1
v(x) = eˣ
v'(x) = eˣ
u'v + uv' = 1 × eˣ + x × eˣ = eˣ + xeˣ
f'(x) = eˣ + xeˣ + 3
2)
g(x) = (x² + 2x - 1)eˣ
u(x) = x² + 2x - 1
u'(x) = 2x + 2
v(x) = eˣ
v'(x) = eˣ
g'(x) = u'v + uv' = (2x + 2) × eˣ + ( x² + 2x - 1) × eˣ
= 2xeˣ + 2eˣ + x²eˣ + 2xeˣ - eˣ
= x²eˣ + 4xeˣ + eˣ
= eˣ(x² + 4x + 1)
3)
[tex]h(x) = \frac{e^x}{e^x + x}[/tex]
u(x) = eˣ
u'(x) = eˣ
v(x) = eˣ + x
v'(x) = eˣ + 1
[tex]h'(x) = \frac{u'v -uv'}{v^2} = \frac{e^x \times (e^x + x) - e^x \times(e^x +1)}{(e^x+x)^2} = \frac{e^{2x} + xe^x - e^{2x} -e^x}{(e^x+x)^2} = \frac{xe^x -e^x}{(e^x+x)^2} = \frac{e^x(x-1)}{(e^x+x)^2}[/tex]
