CÁLCULO/4. DERIVADAS/1 Algebra de derivadas RA5 2.1 Derivada de un polinomio. Dada la función «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»#f«/mi»«mo»(«/mo»«mi»#x«/mi»«mo»)«/mo»«mo»=«/mo»«mi»#a1«/mi»«msup»«mi»#x«/mi»«mrow»«mi»#n1«/mi»«/mrow»«/msup»«mi»#s1«/mi»«mi»#a2«/mi»«msup»«mi»#x«/mi»«mi»#n2«/mi»«/msup»«mi»#s2«/mi»«mi»#a3«/mi»«msup»«mi»#x«/mi»«mi»#n3«/mi»«/msup»«mi»#s3«/mi»«mi»#n4«/mi»«/math».

Entonces, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»#f«/mi»«mfenced»«mrow»«mi»#x«/mi»«/mrow»«/mfenced»«/mrow»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«/math»

]]> Solución:

Para resolver, aplicamos álgebra de derivadas. En particular, ocuparemos las siguientes propiedades:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»p«/mi»«mfenced»«mrow»«mi»#x«/mi»«/mrow»«/mfenced»«mo»±«/mo»«mi»q«/mi»«mfenced»«mrow»«mi»#x«/mi»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»p«/mi»«mfenced»«mrow»«mi»#x«/mi»«/mrow»«/mfenced»«/mrow»«/mfenced»«mo»±«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»q«/mi»«mfenced»«mrow»«mi»#x«/mi»«/mrow»«/mfenced»«/mrow»«/mfenced»«/math»

(1)

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«mi»a«/mi»«mo»·«/mo»«msup»«mi»#x«/mi»«mi»n«/mi»«/msup»«/mrow»«/mfenced»«mo»=«/mo»«mi»a«/mi»«mo»·«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mfenced»«mrow»«msup»«mi»#x«/mi»«mi»n«/mi»«/msup»«/mrow»«/mfenced»«mo»=«/mo»«mi»a«/mi»«mo»·«/mo»«mi»n«/mi»«mo»·«/mo»«msup»«mi»#x«/mi»«mrow»«mi»n«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«/msup»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§nbsp;«/mo»«mo»§#8704;«/mo»«mi»#x«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«mo»,«/mo»«mo»§nbsp;«/mo»«mo»§#8704;«/mo»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8469;«/mi»«mo»§nbsp;«/mo»«mi»d«/mi»«mi»o«/mi»«mi»n«/mi»«mi»d«/mi»«mi»e«/mi»«mo»§nbsp;«/mo»«mi»a«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»§#8477;«/mi»«/math»

(2)

Así, aplicando la propiedad (1) tenemos:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#a1«/mi»«msup»«mi»#x«/mi»«mi»#n1«/mi»«/msup»«mi»#s1«/mi»«mi»#a2«/mi»«msup»«mi»#x«/mi»«mi»#n2«/mi»«/msup»«mi»#s2«/mi»«mi»#a3«/mi»«msup»«mi»#x«/mi»«mi»#n3«/mi»«/msup»«mi»#s3«/mi»«mi»#n4«/mi»«mo»)«/mo»«mo»=«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#a1«/mi»«msup»«mi»#x«/mi»«mi»#n1«/mi»«/msup»«mo»)«/mo»«mo»§nbsp;«/mo»«mi»#s4«/mi»«mo»§nbsp;«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#a4«/mi»«msup»«mi»#x«/mi»«mi»#n2«/mi»«/msup»«mo»)«/mo»«mo»§nbsp;«/mo»«mi»#s5«/mi»«mo»§nbsp;«/mo»«mfrac»«mi»d«/mi»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»(«/mo»«mi»#a5«/mi»«msup»«mi»#x«/mi»«mi»#n3«/mi»«/msup»«mo»)«/mo»«mo»§nbsp;«/mo»«mi»#s6«/mi»«mo»§nbsp;«/mo»«mfrac»«mrow»«mi»d«/mi»«mo»(«/mo»«mi»#a6«/mi»«mo»)«/mo»«/mrow»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«/math»

Luego, aplicando la propiedad (2) tenemos:

Reduciendo, obtenemos finalmente:

Por lo tanto:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi»d«/mi»«mi»#f«/mi»«/mrow»«mrow»«mi»d«/mi»«mi»#x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»#b1«/mi»«msup»«mi»#x«/mi»«mi»#n5«/mi»«/msup»«mi»#s1«/mi»«mi»#b2«/mi»«msup»«mi»#x«/mi»«mi»#n6«/mi»«/msup»«mi»#s2«/mi»«mi»#b3«/mi»«msup»«mi»#x«/mi»«mi»#n7«/mi»«/msup»«/math»

]]> 1.0000000 1.0000000 0 0 #sol Bien. Continúa de esa manera.]]> #sol2 Revisa el desarrollo para identificar tus errores. ¡Tú puedes!]]> <question><wirisCasSession><session lang="es" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mi>aleatorio</mi><mo>{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>p</mi><mo>}</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>aleatorio</mi><mo>{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>n</mi><mo>}</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cero</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math 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xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>*</mo><msup><mi>x</mi><mn>9</mn></msup><mo>-</mo><mn>16</mn><mo>*</mo><msup><mi>x</mi><mn>7</mn></msup><mo>-</mo><mn>21</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a2</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>9</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a3</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>8</mn></math></output></command><command><input><math 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