Question

calcule f'(x) pela definição

a)f(x)=3x-1

b)f(x)=x3

Answer 1

f(x) - f(x0) / x - x0 = f'(x) 

f(x) = 3x - 1 
3x = f(x) - 1 

x = 1 / 3

f(x) = 3x - 1 
3x = f(x) - 1 

x = 1 / 3 

f'(x) = 3x - 1
d / dx (3x -1) = 3 

Answer 2

Resposta:

Explicação passo-a-passo:

a)

$f'(x)= \lim_{h \to 0}\dfrac{f(x+h)-f(x)}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{3.(x+h)-1-(3x-1)}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{3x+3h-1-3x+1}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{3h}{h}\\\\f'(x)= \lim_{h \to 0}3\\\\f'(x)=3$

b)

$f'(x)= \lim_{h \to 0}\dfrac{(x+h)^{3}-x^{3}}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{x^{3}+3x^{2}h+3xh^{2}+h^{3}-x^{3}}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{3x^{2}h+3xh^{2}+h^{3}}{h}\\\\f'(x)= \lim_{h \to 0}\dfrac{h.(3x^{2}+3xh+h^{2})}{h}\\\\f'(x)= \lim_{h \to 0}(3x^{2}+3xh+h^{2})\\\\f'(x)=3x^{2}+3x.0+0^{2}\\\\f'(x)=3x^{2}$