Exercice 1

\(f'(x) = \frac{1}{2\sqrt{x-1}} \) \(D_f=[1;+\infty[\) \(D_f'=]1;+\infty[\)

Le nombre dérivée \(f '(a)\) est le coefficient directeur de la droite tangente (si elle existe) à la courbe en \(x=a\). \(f'(a)=\lim\limits_{h \rightarrow 0} \frac{f (a+h) - f (a)}{h}\)

Exercice 2

$$ \begin{array}{lll} f (x) &=& 3 (1-x^3)^4 \\ f'(x)&=& 3\times 4 \times (-3x^2)(1-x^3)^3 \\ &=&-36 (1-x^3)^3 \\ \end{array} $$

$$ \begin{array}{lll} g (x) &=& \frac{3}{(x+3)^4}\\ g'(x)&=& \frac{-3\times 4}{(x+3)^5}\\ &=& \frac{-12}{(x+3)^5} \end{array} $$

$$ \begin{array}{lll} h (x) &=& x^2 \sqrt{x-1}\\ h'(x)&=& 2 x \sqrt{x-1} + x^2\times \frac{1}{2\sqrt{x-1}}\\ &=& \frac{2 x \sqrt{x-1}\times 2 \sqrt{x-1} + x^2 }{2\sqrt{x-1}}\\ &=& \frac{4 x (x-1) + x^2 }{2\sqrt{x-1}}\\ &=& \frac{4 x^2 - 4x + x^2 }{2\sqrt{x-1}}\\ &=& \frac{5 x^2 - 4x}{2\sqrt{x-1}}\\ &=& \frac{x (5 x - 4)}{2\sqrt{x-1}} \end{array} $$

$$ \begin{array}{lll} k (x) &=& \sqrt{ x - \frac{4}{x}} \\ k' (x) &=& \frac{1+\frac{4}{x^2}}{2\sqrt{x-\frac{4}{x}}} \\ &=& \frac{x^2+4}{2x^2\sqrt{x-\frac{4}{x}}} \\ &=& \frac{x^2+4}{2x^2\sqrt{\frac{x^2-4}{x}}} \\ &=& \frac{(x^2+4)\sqrt{x}}{2x^2\sqrt{x^2-4}} \\ \end{array} $$

Exercice 3 \(f (x) = 16 x - 4 + \frac{2}{2x-6}\) $$ \begin{array}{lll} f' (x) &=& 16 - \frac{4}{(2x-6)^2}\\ &=& 4^2 - \left ( \frac{2}{2x-6} \right)^2\\ &=& \left ( 4 - \frac{2}{2x-6} \right)\left ( 4 + \frac{2}{2x-6} \right) \text{identité remarquable}\\ &=& \frac{(8x-24-2)(8x-24+2)}{(2x-6)^2}\\ &=& \frac{(8x-26)(8x-22)}{(2x-6)^2}\\ &=& \frac{4 (4x-13)(4x-11)}{(2x-6)^2}\\ \end{array} $$ $$ \begin{array}{c|lcccr|} x & -\infty & & 11/4 & & 3 & & 13/4 & & +\infty \\ \hline 4x-13 & & - & & - & & - & 0 & + & \\ \hline 4x+11 & & - & 0 & + & & + & & + & \\ \hline (2x-6)^2 & & + & & + & 0 & + & & + & \\ \hline f'(x) & & + & 0 & - & || & - & 0 & + & \\ \hline f (x) & & \nearrow & & \searrow & || & \searrow & & \nearrow & \\ \hline \end{array} $$

Exercice 1

Le nombre dérivée \(f '(a)\) est le coefficient directeur de la droite tangente (si elle existe) à la courbe en \(x=a\). \(f'(a)=\lim\limits_{h \rightarrow 0} \frac{f (a+h) - f (a)}{h}\)

\(f'(x) = \frac{1}{2\sqrt{x-3}} \) \(D_f=[3;+\infty[\) \(D_f'=]3;+\infty[\)

Exercice 2

$$ \begin{array}{lll} f (x) &=& \frac{3}{(x-2)^3}\\ f'(x)&=& \frac{3\times (-3)}{(x-3)^4}\\ &=& \frac{-9}{(x-3)^4} \end{array} $$

$$ \begin{array}{lll} g (x) &=& x \sqrt{x-3}\\ g'(x)&=& \sqrt{x-3} + x \times \frac{1}{2\sqrt{x-3}}\\ &=& \frac{2\sqrt{x-3}\sqrt{x-3} + x }{2\sqrt{x-3}}\\ &=& \frac{2 (x-3) + x }{2\sqrt{x-3}}\\ &=& \frac{3x-6}{2\sqrt{x-3}} \end{array} $$

$$ \begin{array}{lll} h (x) &=& 2 (1-x^2)^3 \\ h'(x)&=& 2\times 3 \times (-2x)(1-x^2)^2 \\ &=&-12 x (1-x^2)^2 \\ \end{array} $$

$$ \begin{array}{lll} k (x) &=& \sqrt{ 9 x - \frac{3}{x}}\\ k' (x) &=& \frac{9+\frac{3}{x^2}}{2\sqrt{9x-\frac{3}{x}}}\\ &=& \frac{9x^2+3}{2x^2\sqrt{9x-\frac{3}{x}}} \\ &=& \frac{9x^2+3}{2x^2\sqrt{\frac{9x^2-3}{x}}} \\ &=& \frac{(9x^2+3)\sqrt{x}}{2x^2\sqrt{9x^2-3}} \end{array} $$

Exercice 3 \(f (x) = 4x + 1 + \frac{3}{3x-6}\) $$ \begin{array}{lll} f' (x) &=& 4 - \frac{9}{(3x-6)^2}\\ &=& 2^2 - \left ( \frac{3}{3x-6} \right)^2\\ &=& \left ( 2 - \frac{3}{3x-6} \right)\left ( 2 + \frac{3}{3x-6} \right) \text{identité remarquable}\\ &=& \left ( \frac{6x-12-3}{3x-6} \right)\left (\frac{6x-12+3}{3x-6} \right)\\ &=& \frac{(6x-15)(6x-9)}{(3x-6)^2}\\ &=& \frac{9 (2x-5)(2x-3)}{(3x-6)^2} \end{array} $$ $$ \begin{array}{c|lcccr|} x & -\infty & & 3/2 & & 2 & & 5/2 & & +\infty \\ \hline 6x-15 & & - & & - & & - & 0 & + & \\ \hline 6x-9 & & - & 0 & + & & + & & + & \\ \hline (3x-6)^2 & & + & & + & 0 & + & & + & \\ \hline f'(x) & & + & 0 & - & || & - & 0 & + & \\ \hline f (x) & & \nearrow & & \searrow & || & \searrow & & \nearrow & \\ \hline \end{array} $$