Solved example of definition of derivative
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
Expand $\left(x+h\right)^2$
Expand the fraction $\frac{h^2+2xh}{h}$ into $2$ simpler fractions with common denominator $h$
Simplify the fraction $\frac{2xh}{h}$ by $h$
Simplify the fraction by $h$
Simplify
Evaluate the limit $\lim_{h\to0}\left(h+2x\right)$ by replacing all occurrences of $h$ by $0$
$x+0=x$, where $x$ is any expression
Simplifying, we get
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