Derivative Calculator

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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



More References and Links

derivative
rules
formulas
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