# 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

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

## More References and Links

derivativerules

formulas