# Derivative Calculator



A step by step derivative of functions calculator is presented including suggetsed activities.
This calculator uses the rules and formulas to calculate the derivatives.

## Definition and Rules of Derivative

The derivative of a given function $f(x)$ is defined as $f'(x) = \lim_{h\to\ 0} \dfrac{f(x+h)-f(x)}{h}$
Some of the most important rules used by this calculator are listed below.
1 - For $f(x) = x^r$ where r is a constant,     $f'(x) = r x^{r-1}$
2 - For $f(x) = c g(x)$ , where $c$ is a constant,    $f'(x) = c g'(x)$
3 - For $f(x) = h(x) + g(x)$ ,    $f'(x) = h'(x) + g'(x)$
4 - For $f(x) = h(x) \cdot g(x)$ ,    $f'(x) = h'(x) \cdot g(x) + h(x) \cdot g'(x)$

5 - For $f(x) = \dfrac{ h(x) }{ g(x) }$ ,    $f'(x) = \dfrac{ h'(x) \cdot g(x) - h(x) \cdot g'(x)}{ (g(x))^2}$

## Use of the Derivative Calculator

1 - Enter and edit function $f(x)$ and click "Enter Function" then check what you have entered.
Note that the five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example: f(x) = x^3 - 2*x + 3*cos(3x-3) + e^(-4*x)).(more notes on editing functions are located below)
2 - Click "Calculate Derivative" to obain the first derivative $\displaystyle f'(x)$.
3 - Note that the natural logarirthm is entered as $log(x)$, the natural exponential as   $exp(x)$.
4 - Note that a function $f(x)$ to some power $n$ is entered as: $(f(x))^n$. Example:   $sin^2(2x-1)$   is entered as   (sin(2x-1))^2.
5 - Note that the derivative is first given by applying the . rules of derivatives and a second form in simplified form which you may have to further simplify.
6 - Note Enter decimal numbers as fractions between brackets. Example : enter (1/2) instead of 0.5

$f(x)$ =

Notes: In editing functions, use the following:
1 - The five operators used are: + (plus) , - (minus), / (division) , ^ (power) and * (multiplication). (example:    f(x) = 2*x^3 + 3*cos(2x - 5) + log(x)  )
2 - The function square root function is written as (sqrt). (example: sqrt(x^2-1) for $\sqrt {x^2 - 1}$ )
3 - The exponential function is written as exp(x). (Example: exp(2*x+2)    for    $e^{2*x+2}$ )
4 - The log base e function is written as log(x). (Example: log(x^2-2)    for    $ln(x^2 - 2$ )
Here are some examples of functions that you may copy and paste to practice:
x^2 + 2x - 3       (x^2+2x-1)/(x-1)       1/(x-2)       log(2*x - 2)      sqrt(x^2-1)
2*sin(2x-2)       exp(2x-3)       (2*sin(2x-1))^2       (x-1)(x+3)^3

## Suggested Activities

We suggest using this calculator to check the 5 rules given above.
1 - Enter f(x) = x^6 and check rule (1) above
2 - Enter f(x) = 2(x^3) and check rule (2) above
3 - Enter f(x) = x^2 + x^5 and check rule (3) above
4 - Enter f(x) = (x-2)(x+3) and check rule (4) above
5 - Enter f(x) = (x-2)/(x+3) and check rule (5) above