I've some equation about "Derivatives" to ask about. Please, show me how to do that step by step: $$f(x) = \frac{3x^2+1}{2}.$$ $f'(x)= ?$
Derivative Calculation
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1"you people" can seem somewhat dismissive or derogatory, so I took the liberty of changing it. – 2012-03-06
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0Do you know all the properties of the operation, "taking derivative" ? – 2012-03-06
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0Yes sir, it's " f`(x)= ? " – 2012-03-06
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0What do you know about derivatives? Do you know/are you allowed to use some (which?) basic properties of derivatives? Some formulas? Do you need to compute $f'(x)$ using the limit definition? – 2012-03-06
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0@KannappanSampath: I've been studying "Derivatives" since last morning, so, I don't know that much about it for now... Any help, please? – 2012-03-06
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0@ArturoMagidin: Sorry about my English. Most of the time, I'm trying my best here actually... – 2012-03-06
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0@ArturoMagidin: Well, I know some basic rules about it, how to add, how to subtract, how to multiply, how to divide but, for this time, I'm just a beginner about it... – 2012-03-06
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0@ArturoMagidin: Sure, just go ahead, you've the access to edit my posts also. Besides, it's a good behaviour or habit to that in a useful way, right. So, thank you... :) – 2012-03-06
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0:) Thank you guys, now I'm having a litte affair to choose more prior, maybe a better answer from all these replies... :) Thanks... – 2012-03-06
5 Answers
Easy:$\frac{d}{dx} {\frac{3}{2}x^{2}+\frac{1}{2}} = 3x.$ Just do some algebra, and then use the Sum Rule: $\frac{d(f+g)}{dx} = \frac{df}{dx} + \frac{dg}{dx}$ followed by the Power Rule: $\frac{d}{dx} x^{n} = nx^{n-1}$. Remember that you don't have to differentiate coefficients, just leave them out of the derivative.
Each step should follow one of the derivative rules that you know about. The notation "$\frac{d}{dx}$" in what follows means "take the derivative of".
$$ \begin{align*} f'(x) &= \frac{d}{dx}\left[ \frac{3x^2 + 1}{2} \right]\\ &= \frac{d}{dx}\left[ \frac{1}{2}(3x^2 + 1) \right], \quad\textrm{(algebra)}\\ &= \frac{1}{2}\frac{d}{dx}\left[3x^2 + 1 \right], \quad\textrm{(constant multiple rule)}\\ &= \frac{1}{2}\left( \frac{d}{dx}[3x^2] + \frac{d}{dx}[1] \right), \quad \textrm{(sum/difference rule)}\\ &= \frac{1}{2}\left( 3\frac{d}{dx}[x^2] + \frac{d}{dx}[1] \right), \quad \textrm{(constant mult. rule again)}\\ &= \frac{1}{2}\left( 3(2x) + \frac{d}{dx}[1] \right), \quad \textrm{(power rule)}\\ &= \frac{1}{2}\left( 3(2x) + 0 \right), \quad \textrm{(derivative of a constant is 0 -- really just power rule)}\\ &= 3x, \quad \textrm{(algebra to simplify answer)} \end{align*} $$ Now as you do more and more of these problems, you'll find which steps you can do in your head, until you get to the point where it becomes a one-line problem!
Hope this helps!
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0An other awesome help here, thanks Shaun... :) – 2012-03-06
It's simple: Just apply the definition of the derivative ($f$ is a polynomial so is differentiable, which we can prove).
$f'(x) = \lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$
$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h} = \lim \limits_{h \to 0} \frac{(3(x+h)^2 + 1) -(3x^2 +1)}{2h} = $ . . . ?
All it takes is a little manipulation. You should find some very important things will cancel out and the limit will be easy to take.
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0Well, I'm not skilled enough to understand your very valuable reply here for now, but thank you very much Tyler... :) This might work for others at least... :) – 2012-03-06
$\frac{1}{2}\frac{d}{dx}(3x^2 + 1) = \frac{1}{2}(\frac{d}{dx}3x^2 + \frac{d}{dx}1) = \frac{1}{2}(3\frac{d}{dx}x^2 + \frac{d}{dx}1)$
Since 1 is a constant its derivative becomes 0 and as for $x^2$ we have a rule that states that if $f(x) = x^r$ then $f'(x) = r\cdot x^{r-1}$. With that in mind we get
$\frac{1}{2}(3\frac{d}{dx}x^2 + \frac{d}{dx}1) = \frac{1}{2}(3(2x) + 0) = \frac{1}{2}6x = 3x$
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0I understand it now, thanks sir... :) – 2012-03-06
$f(x) = \frac32 x^2 + \frac12$. Just use the Power Rule.
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0Humm... thanks Patrick... – 2012-03-06