Solution :
\( f'(x) = 2 \times 3x^2 – 5 \times 2x + 3 = 6x^2 – 10x + 3 \)

Solution :
\( g'(x) = \frac{1}{2\sqrt{4x + 1}} \times 4 = \frac{2}{\sqrt{4x + 1}} \)

Solution :
\( h'(x) = \frac{(2)(x^2 + 3) – (2x – 1)(2x)}{(x^2 + 3)^2} \)
\( = \frac{2x^2 + 6 – 4x^2 + 2x}{(x^2 + 3)^2} = \frac{-2x^2 + 2x + 6}{(x^2 + 3)^2} \)

Solution :
1. \( f(2) = 4 – 6 = -2 \)
2. \( f'(x) = 2x – 3 \) ⇒ \( f'(2) = 1 \)
3. Équation : \( y = 1(x – 2) – 2 = x – 4 \)

Solution :
\( f'(1) = \lim_{h\to 0} \frac{f(1+h)-f(1)}{h} \)
\( = \lim_{h\to 0} \frac{\frac{(1+h)^2+1}{1+h} – 2}{h} = \lim_{h\to 0} \frac{h^2 + 2h + 2 – 2 – 2h}{h(1+h)} \)
\( = \lim_{h\to 0} \frac{h^2}{h(1+h)} = 0 \)

Solution :
1. \( f'(x) = 3x^2 – 12x + 9 \)
2. Résoudre \( 3x^2 – 12x + 9 = 0 \) ⇒ \( x = 1 \) ou \( x = 3 \)
3. Tableau de signes :
• \( f’ > 0 \) sur \( ]-\infty,1[ \) et \( ]3,+\infty[ \) ⇒ f croissante
• \( f’ < 0 \) sur \( ]1,3[ \) ⇒ f décroissante

Solution :
\( k'(x) = 5(3x^2 – 2)^4 \times 6x = 30x(3x^2 – 2)^4 \)

Solution :
La vitesse est \( v(t) = x'(t) = 3t^2 – 18t + 24 \)
Résoudre \( 3t^2 – 18t + 24 = 0 \) ⇒ \( t = 2 \) s et \( t = 4 \) s

Solution :
1. \( f(x) = x^{3/2} \)
2. \( f'(x) = \frac{3}{2}x^{1/2} \)
3. \( f”(x) = \frac{3}{4}x^{-1/2} = \frac{3}{4\sqrt{x}} \)

Solution :
1. \( f(x) = x + 1 + \frac{1}{x} \) ⇒ \( f'(x) = 1 – \frac{1}{x^2} \)
2. \( f'(x) > 0 \) ⇔ \( x^2 > 1 \) ⇔ \( x \in ]-\infty,-1[ \cup ]1,+\infty[ \)
3. Variations :
• Croissante sur \( ]-\infty,-1[ \) et \( ]1,+\infty[ \)
• Décroissante sur \( ]-1,0[ \) et \( ]0,1[ \)