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Definition of Derivative Calculator

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1

Solved example of definition of derivative

$derivdef\left(x^2\right)$
2

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

$\lim_{h\to0}\left(\frac{\left(x+h\right)^2-x^2}{h}\right)$
3

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

$\lim_{h\to0}\left(\frac{2xh+h^2}{h}\right)$
4

Factor the polynomial $2xh+h^2$ by it's GCF: $h$

$\lim_{h\to0}\left(\frac{h\left(2x+h\right)}{h}\right)$
5

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

$\lim_{h\to0}\left(2x+h\right)$
6

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\lim_{h\to0}\left(2x\right)+\lim_{h\to0}\left(h\right)$
7

The limit of a constant is just the constant

$2x+\lim_{h\to0}\left(h\right)$
8

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

0
9

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

$2x$

Final Answer

$2x$

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