** Final Answer

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** Step-by-step Solution **

Problem to solve:

** Specify the solving method

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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

Square of the first term: $\left(x\right)^2 = [a^2]$.

Double product of the first by the second: $2\left(x\right)\left(h\right) = [2ab]$.

Square of the second term: $\left(h\right)^2 = [b^2]$.

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Expand $\left(x+h\right)^2$

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Cancel like terms $x^2$ and $-x^2$

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Factor the polynomial $2xh+h^2$ by it's greatest common factor (GCF): $h$

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Simplify the fraction $\frac{h\left(2x+h\right)}{h}$ by $h$

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Evaluate the limit $\lim_{h\to0}\left(2x+h\right)$ by replacing all occurrences of $h$ by $0$

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$x+0=x$, where $x$ is any expression

** Final Answer

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