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Derivada de xn
(Matemática | Cálculo | Derivadas | Tabla de | xn)
(d/dx) xn = n x(n-1)



Tres demostraciones

Demostración de (d/dx) xn : desde (d/dx) e(n ln x)

Dando: (d/dx) ex = ex; (d/dx) ln(x) = 1/x; La Regla de la cadena.
Resuelva:

(d/dx) xn = (d/dx) e(n ln x)
= (d/du) eu (d/dx) (n ln x) (Fije u = n ln x)
= [e(n ln x)] [n/x] = x^n n/x = n x(n-1)     Q.E.D.

Demostración de (d/dx) xn : desde la Integral

Given: (integral)xn dx = x(n+1)/(n+1) + c; El teorema fundamental de cálculo.
Resuelva:

(integral)x(n-1) dx = xn / n
(d/dx) xn / n = (d/dx)(integral)x(n-1) dx = x(n-1)
1/n (d/dx) xn = x(n-1)
(d/dx) xn = n x(n-1)     Q.E.D.

Demostración de (d/dx) xn : algebraico

Dando: (a+b)n = (n, 0) an b0 + (n, 1) a(n-1) b1 + (n, 2) a(n-2) b2 + .. + (n, n) a0 bn
Aquí (n,k) es el coeficiente binómio = n! / ( k! (n-k)! )

Resuelva:

(d/dx) xn = lim(d->0) ((x+d)n - xn)/d
= lim [ xn + (n, 1) x(n-1) d + (n, 2) x(n-2) d2 + .. + x0 dn - xn ] / d
= lim [ (n,1) x(n-1) d + (n, 2) x(n-2) d2 + .. + x0 dn ] / d
= lim (n,1) x(n-1) + (n, 2) x(n-2) d + (n, 3) x(n-3) d2 + .. + x0 dn
= lim (n, 1) x(n-1) (todos los téminos derechos se cancelan a causa de el factor d)
= lim (n, 1) x(n-1) = n! / ( 1! (n-1)! ) x(n-1) = n x(n-1)     Q.E.D.

  
 
  

 
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