Réponse :
Bonjour
Exercice 5
1) F(x) = x² + 1/x²
F'(x) = 2x - 2x/x² = 2x - 2/x³ = (2x⁴ - 2)/x³ = 2(x⁴ - 1)/x³ = f(x)
2) F(x) = x - ln(1 + [tex]e^{x}[/tex])
F'(x) = 1 - [tex]e^{x}[/tex]/(1 + [tex]e^{x}[/tex]) = (1 + [tex]e^{x}-e^{x}[/tex])/(1 + [tex]e^{x}[/tex]) = 1/(1 + [tex]e^{x}[/tex]) = f(x)
3) F(x) = ln(lnx)
F'(x) = (1/x)/lnx = 1/xlnx = f(x)
4) F(x) = x cosx
F'(x) = 1×cosx + x(-sinx) = cosx - x sinx = f(x)
Exercice 7
1) f(x) = x⁴ - 4x³ + x² - 4x + 3
F(x) = x⁵/5 - x⁴ + x³/3 - 2x² + 3x
2) f(x) = (x² - 2x + 1)/3
F(x) = (x³/3 - x² +x)/3
3) f(x) = 1 - 1/x³
F(x) = x + 1/(2x²)
4) f(x) = -1/x³ + 4/x² - 1
F(x) = 1/(2x²) - 4/x - x
5) f(x) = 4/x + 2[tex]e^{x}[/tex]
F(x) = 4ln(x) + 2[tex]e^{x}[/tex]
