How do I find the derivative of $f (x) = x^{3}$ $f(x)=x^3$ from first principles?

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Answer 1 · 2014-10-11T01:29:14

First Principles $\to$ $->$ Difference Quotient

$f'(x)=lim_(h->0)(f(x+h)-f(x))/h$

$f(x)=x^3$

$f(x+h)=(x+h)^3$

$f'(x)=lim_(h->0)((x+h)^3-x^3)/h$

$f'(x)=lim_(h->0)((x+h)(x^2+2xh+h^2)-x^3)/h$

$f'(x)=lim_(h->0)(x^3+2x^2h+xh^2+x^2h+2xh^2+h^3-x^3)/h$

$f'(x)=lim_(h->0)(2x^2h+xh^2+x^2h+2xh^2+h^3)/h$

$f'(x)=lim_(h->0)(h*(2x^2+xh+x^2+2xh+h^2))/h$

$f'(x)=lim_(h->0)2x^2+xh+x^2+2xh+h^2$

$f'(x)=lim_(h->0)3x^2+xh+2xh+h^2$

$f'(x)=3x^2+x(0)+2x(0)+(0)^2$

$f'(x)=3x^2$

How do I find the derivative of f(x)=x^3f(x)=x3f(x)=x^3 from first principles?