Utilice la definición de derivada para hallar la derivada de la siguiente función

1. Utilice la definición de derivada para hallar la derivada de la siguiente función

a) f(x)= 5x^2

Lim f(x+h) – f(x) = 5 ( x + h )^2 – 5x^2 = 5 (x^2 + 2xh + h^2) – 5x^2

h h h h

Lim 5x^2+ 10xh + 5 h^2– 5x^2 = 10xh + 5 h^2 = h (10x + 5h) =

h h h h

Lim 10x + 5(0) = 10x f ’(x) = 10x

h

b) f(x) = t – 1 .

t^2 + 2t +1

Lim f(t+h) – f(t) = t+h – 1 – t – 1 .

h h (t+h)^2+ 2 (t + h) + 1 t^2 + 2t +1

Lim t+h – 1_______ – t – 1 .

h (t+h)^2+ 2 (t + h) + 1 t^2 + 2t +1

h

Lim ( t+h – 1)( t^2 + 2t +1) – (t^2+2th+h^2+2t + 2h +1)(t-1)

h (t^2+2th+h^2+2t + 2h +1) (t^2+2t+1) h

Lim (t^3 + 2t^2+t+ht^2+2th + h – t^2 – 2t – 1) – (t^3-t^2 + 2t^2 h-2th+ h^2 t-h^2+2t^2 - 2t + 2ht – 2h +t -1)

h (t^2+2th+h^2+2t + 2h +1) (t^2+2t+1) h

Lim t^3 + 2t^2+t+ht^2+2th + h – t^2 – 2t – 1 – t^3+t^2 -2t^2 h+2th- h^2 t+h^2-2t^2+ 2t - 2ht +2h -t +1

h (t^2+2th+h^2+2t + 2h +1) (t^2+2t+1) h

Lim ht^2+2th + h -2t^2 h+2th- h^2 t+h^2- 2ht +2h

h (t^2+2th+h^2+2t + 2h +1) (t^2+2t+1)

h

Lim h (ht^2+2th + h -2t^2 h+2th- h^2 t+h^2- 2ht +2h) = h(t^2+ 2t+1-2t^2+ 2t-ht+h-2t+2)

h h (t^2+2th+h^2+2t + 2h +1) (t^2+2t+1) h(t^2+2th+h^2+2t + 2h +1) (t^2+2t+1)

Lim t^2+ 2t+1-2t^2+2 = 〖 t〗^2+ 2t+3

h (t^2+2t +1) (t^2+2t+1) (t^2+2t+1)^2

f ’(x) = 〖 t〗^2+ 2t+3

(t^2+2t+1)^2

2. Determine la derivada de las siguientes funciones:

a) f(x) =x^2 = f’(x) =2x

b) f(x) =x^5 = f’(x) =5x^4

c) f(x) =2 = f’(x) = 0

d) f(x) =√x = f’(x) = 1/(2√x )

e) f(x) =∛x = f’(x) = 1/(3∛(x^2 ) )

f) f(x) =e^x = f’(x)

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