Mathematics
OpenStudy (anonymous):

f(x) = x^3 - x; using increment form of the definition of derivative, find f'(x).

OpenStudy (anonymous):

as in $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$??

OpenStudy (anonymous):

i am not familiar with the term "increment form"

OpenStudy (anonymous):

yes that one pretty sure

OpenStudy (anonymous):

OpenStudy (anonymous):

ok lets to it

OpenStudy (anonymous):

$f(x)=x^3-x$ $f(x+h)=(x+h)^3-(x+h)$ then algebra $f(x+h)=(x+h)^3-(x+h)=x^3+3x^2h+3xh^2+h^2-x-x$

OpenStudy (anonymous):

damn typo $f(x+h)=(x+h)^3-(x+h)=x^3+3x^2h+3xh^2+h^2-x-h$

OpenStudy (anonymous):

now $f(x+h)-f(x)=x^3+3x^2h+3xh^2+h^2-x-h-(x^3-x)$ $=3x^2h+3xh^2+h^3-h$

OpenStudy (anonymous):

and finally $\frac{f(x+h)-f(x)}{h}=\frac{=3x^2h+3xh^2+h^3-h}{h}=\frac{h(3x^2+3xh+h^2-1)}{h}$ $=3x^2+3xh+h^2-1$ now take the limit as h goes to zero (be replacing h by zero) and you get $3x^2-1$ just like it should be

OpenStudy (anonymous):

hope all the steps are there i see the binary dust is coming, so i will go watch judge judy