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

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