JORGEMARIOMEDINAMARTIN

1)Derivar:

$\ y(x)=3x^{3}-4x^{2}+5x+16$

$\ y'(x)=9x^{2}-8x+5+0$

$\ y'(x)=9x^{2}-8x+5$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

2)Hallar el limite:

$\lim _{x\to 1}{\frac {x^{2}-1}{x-1}}$

$\lim _{x\to 1}{\frac {(x+1)(x-1)}{x-1}}$

$\lim _{x\to 1}{x+1}$

$\lim _{x\to 1}{1+1}$

$\ 2$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

3)Hallar el limite:

$\lim _{x\to 1}{\frac {2x^{2}-5x-3}{x-3}}$

$\lim _{x\to 3}{\frac {(2x+1)(x-3)}{(x-3)}}$

$\lim _{x\to 3}{2x+1}$

$\lim _{x\to 3}{2(3)+1}$

$\lim _{x\to 3}{6+1}$

$\ 7$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

4)Con la formula de derivada en limite halle:

$\lim _{h\to 0}{2x^{2}}$

$\lim _{h\to 0}{\frac {2(x+h)^{2}-2x^{2}}{h}}$

$\lim _{h\to 0}{\frac {2(x^{2}+2xh+h^{2})-2x^{2}}{h}}$

$\lim _{h\to 0}{\frac {2x^{2}+4xh+2h^{2})-2x^{2}}{h}}$

$\lim _{h\to 0}{\frac {2x^{2}+4xh+2h^{2}-2x^{2}}{h}}$

$\lim _{h\to 0}{\frac {4xh+2h^{2}}{h}}$

$\lim _{h\to 0}{\frac {h(4x+2h)}{h}}$

$\lim _{h\to 0}{4x+2h}$

$\ 4x$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

5)Con la formula de derivada en limite halle:

$\lim _{h\to 0}{senx}$

$\lim _{h\to 0}{\frac {sen(x+h)-senx}{h}}$

$\lim _{h\to 0}{\frac {(senxcosh+senhcosx)-senx}{h}}$

$\lim _{h\to 0}{\frac {senx(cosh-1)+senhcosx}{h}}$

$\lim _{h\to 0}{\frac {senx(cosh-1)}{h}}+{\frac {senhcosx}{h}}$

$\lim _{h\to 0}senx{\frac {(cosh-1)}{h}}+cosx{\frac {senh}{h}}$

$\lim _{h\to 0}senx(0)+cosx(1)$

$\ cosx$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

6)Con la formula de derivada en limite halle:

$\lim _{h\to 0}{12x^{2}-6x+6}$

$\lim _{h\to 0}{\frac {(12(x+h)^{2}-6(x+h)+6)-(12x^{2}-6x+6)}{h}}$

$\lim _{h\to 0}{\frac {(12(x^{2}+2xh+h^{2})-6x-6h+6-12x^{2}+6x-6}{h}}$

$\lim _{h\to 0}{\frac {12x^{2}+24xh+12h^{2}-6x-6h+6-12x^{2}+6x-6}{h}}$

$\lim _{h\to 0}{\frac {24xh+12h^{2}-6h}{h}}$

$\lim _{h\to 0}{\frac {h(24x+12h-6)}{h}}$

$\lim _{h\to 0}{24x+12h-6}$

$\ 24x-6$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

7)Hallar las derivadas en su maxima expresion de:

$\ y=x^{4}-x^{3}+3x^{2}$

$\ y''=4x^{3}-3x^{2}+6x$

$\ y'''=12x^{2}-6x+6$

$\ y''''=24x-6$

$\ y'''''=24$

$\ y''''''=0$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

8)Hallar la derivada de:

$\ y={\frac {x^{4}-1}{x^{2}}}$

$\ y'={\frac {(x^{2})(4x^{3})-(4x^{3}-1)(2x)}{(x^{2})^{2}}}$

$\ y'={\frac {(4x^{5})-(8x^{4}-2x)}{(x^{2})^{2}}}$

$\ y'={\frac {4x^{5}-8x^{4}+2x}{x^{4}}}$

JORGE MARIO MEDINA MARTIN 5:10, 13 Jul 2004

100)$\lim _{x\to 1}{\frac {{\sqrt[{2}]{x}}-1}{{\sqrt[{3}]{x}}-1}}$

$\lim _{x\to 1}{\frac {({\sqrt[{2}]{x}}-1)({\sqrt[{2}]{x}}+1)}{({\sqrt[{3}]{x}}-1)({\sqrt[{2}]{x}}+1)}}$

$\lim _{x\to 1}{\frac {({\sqrt[{2}]{x}})^{2}-(1)^{2}}{({\sqrt[{3}]{x}}-1)({\sqrt[{2}]{x}}+1)}}$