Find the derivative of $x^2$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x^2$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
Expand the expression $\left(x+h\right)^2$ using the square of a binomial: $(a+b)^2=a^2+2ab+b^2$
Cancel like terms $x^{2}$ and $-x^2$
Expand the fraction $\frac{2xh+h^{2}}{h}$ into $2$ simpler fractions with common denominator $h$
Simplify the resulting fractions
Evaluate the limit $\lim_{h\to0}\left(2x+h\right)$ by replacing all occurrences of $h$ by $0$
$x+0=x$, where $x$ is any expression
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