Bonjour,
1)
[tex]f(x) = \dfrac{x}{2}+1 = \dfrac{1}{2}x+1\\\\\rightarrow f'(x) = \dfrac{1}{2}[/tex]
2)
[tex]f(x)=\dfrac{1}{3}x^3-\dfrac{x^2}{2}+\dfrac{1}{4}x-7=\dfrac{1}{3}x^3-\dfrac{1}{2}x^2+\dfrac{1}{4}x-7\\\\ f'(x)=\dfrac{1}{3}\times 3x^2-\dfrac{1}{2}\times 2x+\dfrac{1}{4} =x^2-x+\dfrac{1}{4}[/tex]
3)
[tex]f(x)=-2x+1+\dfrac{3}{x+4}\\\\ f'(x) = -2+0+\dfrac{u}{v}\rightarrow\dfrac{u'v-v'u}{v^2}\\\\ u=3\quad u'=0\quad v=x+4\quad v'=1\\\\ f'(x)=-2+\dfrac{0\times(x+4)-3}{(x+4)^2}\\\\ f'(x)=-2+\dfrac{-3}{(x+4)^2}\\\\ f'(x) = \dfrac{2(x+4)^2+3}{(x+4)^2}[/tex]
4)
[tex]f(x)=\dfrac{5}{x}+2\sqrt{x}=5\times\dfrac{1}{x}+2\sqrt{x}\\\\ f'(x)=-5\times\dfrac{1}{x^2}+2\times\dfrac{1}{2\sqrt{x}}\\\\ f'(x)=-\dfrac{5}{x^2}+\dfrac{1}{\sqrt{x}}\\\\ f'(x)=\dfrac{-5\sqrt{x}+x^2}{x^2\sqrt{x}}\\\\ f'(x)=\dfrac{\sqrt{x}(-5\sqrt{x}+x^2)}{x^3}\\\\ f'(x)=\dfrac{-5x+x^2\sqrt{x}}{x^3}\\\\ f'(x)=\dfrac{x(-5+x\sqrt{x}}{x^3}\\\\ f'(x)=\dfrac{-5+x\sqrt{x}}{x^2}[/tex]
5)
[tex]f(x)=\dfrac{1}{1-x}\\\\ f'(x)=\dfrac{1}{u}\rightarrow \dfrac{1}{u^2}\\\\ f'(x)=\dfrac{1}{(1-x)^2} [/tex]
6)
[tex]f(x)=x\sqrt{x}\\\\ f'(x)=u\times v\rightarrow u'v+v'u\\\\ u=x\quad u'=1\quad v=\sqrt{x} \quad v'=\dfrac{1}{2\sqrt{x}}\\\\ f'(x)=\sqrt{x}+\dfrac{1}{2\sqrt{x}}\times x\\\\ f'(x) = \sqrt{x}+\dfrac{x}{2\sqrt{x}}\\\\ f'(x)=\sqrt{x}+\dfrac{\sqrt{x}}{2}\\\\ f'(x)=\dfrac{3}{2}\sqrt{x}[/tex]
7)
[tex]f(x)=3x^2-\dfrac{x+1}{x-1}\\\\ f'(x)=3\times2x-\dfrac{u}{v}\rightarrow\dfrac{u'v-v'u}{v^2}\\\\ u=x-1\quad u'=1\quad v=x+1\quad v'=1\\\\ f'(x)=6x-\dfrac{(x+1)-(x-1)}{(x+1)^2}\\\\ f'(x)=\dfrac{6x(x+1)^2-2}{(x+1)^2}[/tex]
