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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\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