hey!
can anyone help me how to get the derivative of an equation?? any equation! PLEASE show me how! you may give your own example but PLEASE do it step by step so I may understand...
Thanks! That derivative thing is driving me nuts!
hey!
can anyone help me how to get the derivative of an equation?? any equation! PLEASE show me how! you may give your own example but PLEASE do it step by step so I may understand...
Thanks! That derivative thing is driving me nuts!
here's a function.. please show me how..
f(x) = 3x²-2x+4
btw, there's this RULES ON DERIVATIVES, can you give me any shortcut how to familiarize with them easily?? i'm really bad at familiarizing, worse memorizing things.. i really forget thing easily..
The derivative of any function $\displaystyle f(x)$ is
$\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}$.
The calculation of this limit (known as finding the derivative by first principles) can be long and tedious. Thankfully, in many cases the derivative of a function can be found using a rule.
First: The derivative of a sum/difference of functions is the sum/difference of the functions' derivatives.
For a polynomial $\displaystyle f(x) = ax^n$, the derivative $\displaystyle f'(x) = nax^{n-1}$. You could think of this as "multiplying the coefficient by the power, then subtracting $\displaystyle 1$ from the power".
So in your case:
$\displaystyle f(x) = 3x^2 - 2x + 4$
First principles:
$\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$
$\displaystyle = \lim_{h \to 0}\frac{3(x + h)^2 - 2(x + h) + 4 - (3x^2 - 2x + 4)}{h}$
$\displaystyle = \lim_{h \to 0}\frac{3x^2 + 6xh + h^2 - 2x - 2h + 4 - (3x^2 - 2x + 4)}{h}$
$\displaystyle = \lim_{h \to 0}\frac{6xh + h^2 - 2h}{h}$
$\displaystyle = \lim_{h \to 0}(6x + h - 2)$
$\displaystyle = 6x - 2$.
Using the rule: Evaluate the derivative of each term using the rule.
$\displaystyle f(x) = 3x^2 - 2x + 4$
$\displaystyle = 3x^2 - 2x^1 + 4x^0$.
$\displaystyle f'(x) = 2\cdot 3x^{2-1} - 1\cdot 2x^{1-1} + 0\cdot 4x^{0-1}$
$\displaystyle = 6x - 2$.
As you can see, using the rule is much easier than using first principles.