Bonjour,
Question 2 :
[tex]f(t) = (2t+4)e^{-2t}[/tex]
[tex]u = 2t + 4[/tex]
[tex]u ' = 2[/tex]
[tex]v = e^{-2t}\\v' = -2e^{-2t}[/tex]
[tex]f'(t) = u'v + uv'[/tex]
[tex]f'(t) = 2 \times e^{-2t} + (2t + 4) \times -2e^{-2t}[/tex]
[tex]f'(t) = 2e^{-2t} -4te^{-2t} - 8e^{-2t}[/tex]
[tex]f'(t) = -4te^{-2t} - 6e^{-2t}[/tex]
[tex]f'(t) = e^{-2t}(-4t -6)[/tex]
Question 3 :
[tex]g(x) = xe^x[/tex]
[tex]u = x \\u'= 1[/tex]
[tex]v = e^x\\v' = e^x[/tex]
[tex]g'(x) = 1 \times e^x + x \times e^x \\g'(x) = e^x + xe^x \\g'(x) = e^x(1 + x)[/tex]
[tex]y = f'(a)(x-a) + f(a)[/tex]
[tex]f'(1) = e^1 (1 + 1 ) = e \times 2 = 2e \\f(1) = 1 \times e^1 = 1 \times e = e[/tex]
[tex]y = 2e(x - 1) + e \\y = 2ex - 2e + e \\y = 2x - e[/tex]
