Corredorclau

DERIVADAS

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1)Derivar:$F(x)=x^{4}-7x^{2}+6$

$F'(x)=4x^{3}-14x$

--Corredorclau 14:08, 12 Jul 2004 (UTC)

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2)Derivar:$F(x)=5x^{2}-2x-7$

$F'(x)=10x-2$

--Corredorclau 14:15, 12 Jul 2004 (UTC)

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3)Derivar:$F(x)=(2x^{-}5)^{2}$

$F'(x)=2(2x^{2}-5)(4x)$

$F'(x)=(4x^{2}-10)(4x)$

$F'(x)=16x^{3}-40~~~by:corredorclau$

Los siguientes ejercicios fueron tomados del libro de Howard E. Taylor.

4)Derivar:$f(x)={\frac {3}{x}}+{\frac {7}{x^{2}}}-x^{-3}$

$f'(x)={\frac {x-3}{x^{2}}}+{\frac {x^{2}-(2x)(7)}{x^{4}}}+3x^{-4}$

$f'(x)={\frac {-3}{x^{2}}}-{\frac {14x}{(x^{4}}}+3x^{-4}$

$f'(x)={\frac {-3}{x^{2}}}-{\frac {14}{(x^{3}}}+{\frac {3}{x^{4}}}$

Corredorclau 18:18, 13 Jul 2004 (UTC)

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5)Derivar:$F(x)=3x^{\frac {1}{3}}+7senx$

$F'(x)={\frac {1}{3}}+3x^{{\frac {1}{3}}-1}+7cosx$

$F'(x)=3x^{\frac {-2}{3}}+7cosx$

Corredorclau

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6)Derivar:$F(x)=x{\sqrt {5+x^{2}}}$

$F'(x)={\sqrt {5+x^{2}}}+x\cdot {\frac {1}{2{\sqrt {5+x^{2}}}}}\cdot 2x$

$F'(x)={\sqrt {5+x^{2}}}+{\frac {2x^{2}}{2{\sqrt {5+x^{2}}}}}$

$F'(x)={\frac {{\sqrt {5+x^{2}}}+x^{2}}{\sqrt {5+x^{2}}}}$

$F'(x)={\frac {5+x^{2}+x^{2}}{\sqrt {5+x^{2}}}}$

$F'(x)={\frac {5+2x^{2}}{\sqrt {5+x^{2}}}}$

Corredorclau

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7)Derivar:$\ 3x^{2}+4y^{2}=12$

$\ 6x+8yy'=0$

$\ 8yy'=6x$

$\ y'={\frac {6x}{8y}}$

$\ y'={\frac {-3x}{4y}}$

Corredorclau 21:44, 13 Jul 2004 (UTC)

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8)Derivar:$\ F(x)=(x^{2}-4)(x^{4}+5)$

$\ F'(x)=(2x)(x^{4}+5)+(x^{5}-4)(4x^{3})$

$\ F'(x)=2x^{5}+10x+4x^{5}-16x^{3}$

$\ F'(x)=6x^{5}-16x^{3}+10x$

Corredorclau 12:48, 13 Jul 2004 (UTC)

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9)Derivar:$\ F(x)=2x^{2}(2x-4)$

$\ F'(x)=4x(2x-4)+2x^{2}(2)$

$\ F'(x)=8x^{2}-16x+4x^{2}$

$\ F'(x)=12x2-16x$

Corredorclau 12:48, 13 Jul 2004 (UTC)

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10)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

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11):$f(x)={\frac {senx}{x}}$

$f'(x)={\frac {cosx(x)-senx}{x^{2}}}$

$f'(x)={\frac {cosx}{x}}-{\frac {senx}{x^{2}}}$

Corredorclau

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12)Derivar:$f(x)={\frac {2x^{3}+4x}{3senx}}$

$f'(x)={\frac {(6X^{2}-4(3senx))-(2x^{3}-4x)3cosx}{(3senx)^{2}}}$

$f'(x)={\frac {18x^{2}senx-12senx-6x^{3}cosx+12xcosx}{9sen^{2}x}}$

Corredorclau

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13)Derivar:$f(x)={\frac {1}{(x^{2}-2x)^{2}}}$

$f'(x)={\frac {2(x^{2}-2x)(2x-2)}{(x^{2}-2x)^{4}}}$

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

Corredorclau

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14)Derivar:$f(x)={\frac {1-senx}{1+cosx}}$

$f'(x)={\frac {-cosx(1+cosx)-(1-senx)-senx}{1+cosx}}$

$f'(x)={\frac {-cosx-cos^{2}x+senx-sen^{2}x}{1+cos^{2}x}}$

$f'(x)={\frac {-cosx-1+sen^{2}x+senx-sen^{2}x}{(1+cosx)^{2}}}$

$f'(x)={\frac {-cosx-1+senx}{(1+cosx)^{2}}}$

Corredorclau

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15)Derivar:$f(x)={\frac {cos^{2}2x}{x^{2}}}$

$f'(x)={\frac {2(cos2x)sen2x(2)(x^{2})-2x(cos^{2}2x)}{x^{4}}}$

$f'(x)={\frac {4senx-2x}{x^{2}}}$

Corredorclau

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16)Derivar:$f(x)={\frac {cosx}{7}}$

$f'(x)={\frac {-7senx}{49}}$

$f'(x)={\frac {-senx}{7}}$

Corredorclau

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17)Derivar:$f(x)={\frac {(x-1)^{2}}{x^{2}}}$

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

$f'(x)={\frac {(2x-2)(x^{2})-(x^{2}-2x+1)2x}{x^{4}}}$

$f'(x)={\frac {2x^{3}-^{2}x^{2}-2x^{3}+4x^{2}-2x}{x^{4}}}$

$f'(x)={\frac {2x^{2}-2x}{x^{4}}}$

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

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

Corredorclau

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18)Derivar:$f(x)=x+({\frac {1}{x}})^{2}$

$f(x)=2(x+{\frac {1}{x}})(1-{\frac {-1}{x^{2}}})$

$f(x)=(2x+{\frac {2}{x}})(1-({\frac {1}{x^{2}}})$

$f(x)=({\frac {2x^{2}+2}{x}})({\frac {x^{2}-1}{x^{2}}})$

$f(x)={\frac {2x^{2}-2x^{2}+2x^{2}-2}{x^{3}}}$

$f(x)={\frac {2x^{4}-2}{x^{3}}}$

Corredorclau

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19)derivar:$f(x)=cos^{5}(2x^{3}+4)$

$f'(x)=5cos^{4}(2x^{3}+4)(6x^{2})$

$f'(x)=30x^{2}cos^{4}(2x^{3}+4)$

Corredorclau

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20)derivar:$f(x)=sen7x$

$f'(x)=cos7x(7)$

$f'(x)=7cox7x$

Corredorclau

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21)derivar y calcular f'(1):$f(t)=sen(t^{2}+3t+1)$

$f'(t)=cos(t^{2}+3t+1)(2t+3)$

$f'(1)=cos25$

Corredorclau

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22) derivar y hallar g'(1):$g(t)=(t^{2}+9)^{3}(t^{2}-2)^{4}$

$g'(t)=3(t^{2}+9)(2t)(t^{2}-2)^{4}+(t^{2}+9)^{3}(4)(t^{2}-2)^{3}(2t)$

$g'(t)=(3t^{2}+27)^{2}(2t^{3}-4t)^{4}+(t^{2}+9)^{3}(4)(t^{2}-2)^{3}(2t)$

$g'(1)=-113600$

Corredorclau

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23)encontrar la recta tangente de

$y=(x^{2}+1)^{3}(x^{4}+1)^{2}enlospuntos(1,32)$

$y'=3(x^{2}+1)(2x)(x^{4}+1)^{2}+(x^{2}+1)^{3}2(x^{4}+1)(4x^{3})$

$evaluoax=1eny'$

$224$

$laecuaciondelarectaesy=224x-192$

Corredorclau

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24) derivar:$f(x)=5x^{2}+4x+{\frac {7}{x^{5}}}+8cosx$

$f'(x)=10x+4+{\frac {35x^{4}}{x^{1}0}}-8senx$

$f'(x)=10x+4+{\frac {35}{x^{6}}}-8senx$

Corredorclau

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SEGUNDA DERIVADA

25)halle la segunda derivada de:$f(x)=xsenx$

$f'(x)=senx+xcosx$

$f''(x)=cosx+cosx-xsenx$

$f''(x)=2cosx-xsenx$

Corredorclau

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26)halle la segunda derivada de$f(x)=(1+x)^{1}5$

$f'(x)=15(1+x)^{1}4$

$f''(x)=210(1+x)^{1}3$

Corredorclau

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27):$F(x)=(7+x)^{5}$

$f'(x)=5(7+x)^{4}$

$f''(x)=2087+x)^{3}$

Corredorclau

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28):$f(x)=(3-2x)^{5}$

$f'(x)=5(3-2x)^{4}(-2)$

$f'(x)=-10(3-2x)^{4}$

$f''(x)=-40(3-2x)^{3}(-2)$

$f''(x)=80(3-2x)^{3}$

Corredorclau

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29):$f/(x)=(4+2x^{2})^{7}$

$f'(x)=7(4+2x^{2})^{6}(4x)$

$f'(x)=28x(4+2x^{2})^{6}$

$f''(x)=28(4+2x^{2})^{6}+28x(4+2x^{2})^{5}(4x)$

$f''(x)=12(4+2x^{2})^{6}+112x^{2}(4+2x^{2})^{5}$

Corredorclau

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DERIVACION IMPLICITA

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30):$3X^{2}+4Y^{2}=12$

$6X+8YY'=0$

$8YY'=-6X$

$Y'={\frac {6x}{8y}}$

Corredorclau

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31):$x^{3}+y^{3}-3x^{2}+3y^{2}=0$

$3x^{2}+3y^{2}y'-6x+6x+6yy'=0$

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

$y'(3y^{2}+6y)=-3x^{2}+6x$

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

Corredorclau

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32):$senx+cosy=0$

$cosx-senyy'=0$

$-senyy'=-cosx$

$y'={\frac {cosx}{seny}}$

Corredorclau

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33):$4x^{3}+7xy^{2}=2y^{3}$

$12x^{2}+7y^{2}+7x2yy'=6y^{2}y'$

$12x^{2}+7y^{2}=6y^{2}y'-7x2yy'$

$12x^{2}+7y^{2}=y'(6y^{2}-7x2y)$

${\frac {12x^{2}+7y^{2}}{6y^{2}-7x2y}}$

Corredorclau

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DERIVADAS por regla de la cadena

Es utilizada para derivar funciones compuestas,primero se deriva la compuesta y se multiplica por la interna.

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34):$f(t)=({\frac {3t-2}{t+5}})^{3}$

$f'(t)=3({\frac {3t-2}{t+5}})^{2}({\frac {3(t+5)-(3t-2)}{(t+5)^{2}}})$

$f'(t)=3({\frac {3t-2}{t+5}})^{2}({\frac {17}{(t+5)^{2})}}$

Corredorclau

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35):$f(t)={\frac {83t-2)^{3}}{t+5}}$

$f'(t)={\frac {3(3t-2)^{2}3(t+5)-(3t-2)^{3}}{(t+5)^{2}}}$

$f'(t)={\frac {6t+30(3t-2)^{2}-(3t-2)^{3}}{(t+5)^{2}}}$

Corredorclau

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36):$f(x)=sen^{3}x$

$f'(x)=3cosx^{2}(cos)$

Corredorclau

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37):$f(t)=(senttan(t^{2}+1))$

Failed to parse (syntax error): {\displaystyle f´(t)=(costtan(t^2+1)+sentsec^2(t^2+1)(2t))}

$f'(t)=(costtan(t^{2}+1)+sentsec^{2}(2t^{3}+2t))$

Corredorclau

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38):$H(x)={\sqrt {x^{2}-1}}$

$H'(x)={\frac {1}{2{\sqrt {x^{2}-1}}}}(2x)$

$H'(x)={\frac {x}{\sqrt {x^{2}-1}}}$

Corredorclau

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39):$H(x)=sen^{2}x+senx^{2}$

$2senxcosx+cosx^{2}(2x)$

Corredorclau

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MÁXIMOS Y MINIMOS

1) Al lado de un gran muro de una granja. se quiere cercar un terreno que tenga forma rectangular. Se dispone solamente de 100 metros de malla de alambre para construir la cerca para que esta encierre la mayor parte de área posible.

$A=xy$

$2x+y=100$

$y=100-2x$

$A=x(100+2x)$

$A=100x-2x^{2}$

$A'=100-4x$

$A'=0$

$100-4x=0$

$-4x=-100$

$x={\frac {100}{4}}$

$x=25$

$y=100-2x$

$y=100-50$

$y=25$

$A=xy$

$A=25(50)$

$A=1250metros$

Corredorclau

2) cual es la maxima area que puede tener un triangulo rectangulo cuya hipotenusa tenga 5cm de largo.

Por pitagoras

$Z^{2}=x^{2}+y^{2}$

$z=5$

$x^{2}+y^{2}=25$

$y={\sqrt {25-x^{2}}}$

area del triangulo

$A={\frac {xy}{2}}$

$A=x({\frac {\sqrt {25-x^{2}}}{2}})$

${\frac {1}{2}}{\sqrt {25-x^{2}}}+({\frac {(}{x}}{2})({\frac {-2x}{2({\sqrt {25-x^{2}}})}}=0$

${\frac {25-x^{2}-x^{2}}{2{\sqrt {25-x^{2}}}}}=0$

$25-2x^{2}=0$

$x={\sqrt {\frac {25}{2}}}$

$y={\sqrt {25-{\frac {25}{2}}}}$

$y={\sqrt {\frac {25}{2}}}$

$A={\frac {\sqrt {\frac {25}{2}}}{\sqrt {\frac {25}{2}}}}{2}$

$A={\frac {25}{4}}cm^{2}$

Corredorclau