Definition of Derivative Calculator

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

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