Question

Hacer mediante definición de la derivada por favor:(

Answer 1

Definición de la derivada:

$f'(x) = \underset{h \to 0}{Lim} \: \frac{f(x+h) - f(x)}{h}$

Problema B)

$f(x) = 4x^2 +2x -5$

Entonces:

$f'(x) = \underset{h \to 0}{Lim} \: \frac{f(x+h) - f(x)}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{(4(x+h)^2 + 2(x+h) - 5) - (4x^2 +2x -5 )}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{(4(x^2 + 2xh + h^2 ) + 2x+2h - 5) - (4x^2 +2x -5 )}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{ \cancel{4x^2 }+ 8xh + 4h^2 +\cancel{ 2x}+2h - \cancel{5} -\cancel{ 4x^2 } - \cancel{2x} +\cancel{5}}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{ 8xh + 4h^2 +2h }{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{h( 8x + 4h +2) }{h}$

$f'(x) = \underset{h \to 0}{Lim} \: (8x + 4h +2)$

$f'(x) = \: 8x + 4(0) +2$

$f'(x) = \: 8x +2$

Problema C)

$f(x) = 3x^3 + 5x^2 -4x +10$

Entonces:

$f'(x) = \underset{h \to 0}{Lim} \: \frac{f(x+h) - f(x)}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{(3(x+h)^3 + 5(x+h)^2 -4(x+h)+10) - (3x^3 + 5x^2 -4x +10)}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{(3(x^3 + 3x^2 h + 3xh^2 + h^3 ) + 5(x^2 + 2xh + h^2 ) -4x-4h+10) - 3x^3 - 5x^2 +4x -10}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{ \cancel{3x^3 } + 9x^2 h + 9xh^2 +3 h^3 + \cancel{ 5x^2} + 10xh + 5h^2 - \cancel{4x}-4h+\cancel{10} - \cancel{3x^3} -\cancel{ 5x^2 }+ \cancel{4x}- \cancel{10}}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{9x^2 h + 9xh^2 +3 h^3 + 10xh + 5h^2 -4h}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: \frac{h(9x^2 + 9xh +3h^2 + 10x+ 5h -4)}{h}$

$f'(x) = \underset{h \to 0}{Lim} \: (9x^2 + 9xh +3 h^2 + 10x+ 5h -4)$

$f'(x) = \: 9x^2 + 9x(0) +3 (0)^2 + 10x+ 5(0) -4$

$f'(x) = \: 9x^2 + 10x -4$