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 $\left(x+h\right)^2$
Factor the polynomial $h^2+2xh$ by it's GCF: $h$
Simplify the fraction $\frac{h\left(h+2x\right)}{h}$ by $h$
The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$
The limit of a constant is just the constant
Evaluate the limit $\lim_{h\to0}\left(h\right)$ by replacing all occurrences of $h$ by $0$
$x+0=x$, where $x$ is any expression
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