Question

Determina la derivada de las funciones utilizando la definicion por limites

Answer 1

SOLUCION :

 a)  f(x) = -5x+2

    f'(x) = limh→0  [ ( -5*(x +h) + 2 ) - ( -5x+ 2)]/h = lim h→0 -5h/h = -5

 b) f(x) = 2x2 +x

   f'(x) = lim h→0 [ ( 2*(x+h)²+x +h ) - ( 2x²+x )]/h = limh→0 (4xh+2h²+h)/h

  f'(x) = 4x +1

 c) f(x) = 10x³

    f'(x) = lim  h→0  [ 10(x+h)³ -10x³]/h = lim h→0 ( 10x³+30x²h +30xh²+10h³-10x³)/h

  f'(x) = limh→0  h( 30x²+30xh+10h²) /h  = 30x²

  d) f(x)= x³-x+2

      f'(x) = lim h→0  [ (x+h)³- ( x + h)+  2 -x³+x-2 ] /h = lim h→0 ( x³+3x²h +3xh²+h³-x-h +2 -x³+x-2 )/h = lim h→0  h( 3x²+3xh+h²-1)/h = 3x²-1

   e) f(x) = x³-4+x²

    f'(x) = lim h→ 0   [ ( x+h)³- 4 + ( x+h)² - x³+4-x²]/h = lim h→ 0 h*( 3x²+3xh+h²+2x+h )/h

    f'(x) = 3x²+ 2x

    f) f(x) = x/x+2

    f'(x) = lim h→0    [ x+h/x+h+2  - x/x+2 ]/h = lim h→0  [ x²+xh +2x+2h -x²-xh-2x ] h( x+h+2)(x+2) = lim h→0  2/(x+h+2)(x+2) = 2/(x+2)²

   g) f(x)= x²-2x

        f'(x) = lim h→0  [ (x+h)²-2*(x+h) -x²+2x ]/h = lim h→0 ( x²+2xh+h²-2x-2h-x²+2x )/ h = lim h→0  2x+h-2  = 2x-2 .