Final Answer
$2x$
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Step-by-step Solution
Problem to solve:
$derivdef\left(x^2\right)$
1
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
$\lim_{h\to0}\left(\frac{\left(x+h\right)^2-x^2}{h}\right)$
2
Expand $\left(x+h\right)^2$
$\lim_{h\to0}\left(\frac{2xh+h^2}{h}\right)$
3
Expand the fraction $\frac{2xh+h^2}{h}$ into $2$ simpler fractions with common denominator $h$
$\lim_{h\to0}\left(\frac{2xh}{h}+\frac{h^2}{h}\right)$
Simplify the fraction $\frac{2xh}{h}$ by $h$
$\lim_{h\to0}\left(2x+\frac{h^2}{h}\right)$
Simplify the fraction $\frac{h^2}{h}$ by $h$
$\lim_{h\to0}\left(2x+h\right)$
4
Simplify
$\lim_{h\to0}\left(2x+h\right)$
5
Evaluate the limit $\lim_{h\to0}\left(2x+h\right)$ by replacing all occurrences of $h$ by $0$
$2x+0$
$x+0=x$, where $x$ is any expression
$2x$
6
Simplifying, we get
$2x$
Final Answer
$2x$