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# Definition of Derivative Calculator

## Get detailed solutions to your math problems with our Definition of Derivative step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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###  Difficult Problems

1

Here, we show you a step-by-step solved example of definition of derivative. This solution was automatically generated by our smart calculator:

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

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

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

Take the square of the first term: $x$

$x^{2}$

Twice ($2$) the product of the two terms: $x$ and $h$

$2xh$

Take the square of the second term: $h$

$h^{2}$

Add the three results, and we obtain the expanded polynomial

$x^{2}+2xh+h^{2}$
3

Expand the expression $\left(x+h\right)^2$ using the square of a binomial: $(a+b)^2=a^2+2ab+b^2$

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

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

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

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

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

Simplify the fraction $\frac{h^{2}}{h}$ by $h$

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

Simplify the resulting fractions

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

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

$2x+0$
8

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

$2x$

##  Final answer to the problem

$2x$

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