anonymous
  • anonymous
Easy derivative question! x^3-x I know the answer is x^2-1 but I cant get it. I'm doing the limit method. Please show me how to get the correct answer!
Calculus1
katieb
  • katieb
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katieb
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anonymous
  • anonymous
Yes thats the method i was using
anonymous
  • anonymous
i cant get it though
Hero
  • Hero
@jobo What's f(x+h) ?

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anonymous
  • anonymous
I got \[x ^{3}-3x ^{2}\Delta x + 3x \Delta x ^{2} - \Delta x ^{3} -x-\Delta x -x ^{3} +x\]
IrishBoy123
  • IrishBoy123
you should be starting with something like \[\frac{\Delta y}{\Delta x} = \dfrac{\left\{ (x+\Delta x)^3 - (x+\Delta x ) \right\}- \left\{x^3 - x\right\}}{\Delta x}\] which is \[\frac{\Delta y}{\Delta x} = \dfrac{(x+\Delta x)^3 - x^3}{\Delta x} +\dfrac{ -(x+\Delta x) + x}{\Delta x}\] expand that out to: \[\frac{\Delta y}{\Delta x} = \dfrac{(x^3+3 x^2 \Delta x + 3x \Delta x^2 + \Delta x^3) - x^3}{\Delta x} +\dfrac{-(x+\Delta x) + x}{\Delta x}\] that's kinda where you have got to but the wheels seem to have come off.
IrishBoy123
  • IrishBoy123
and the answer is not \(\text{x^2-1}\) :p
SolomonZelman
  • SolomonZelman
You can take the derivative of \("\)x\(^3\)- x\("\), simply by applying the *POWER RULE* to each term in the equation. (I will use this notation to denote the derivative of a function: \(\color{black}{\dfrac{d}{dx}}\) ) \(\color{red}{\rm THE{~~~~}POWER{~~~~}RULE}\) \(\large{\bbox[5pt, lightyellow ,border:2px solid black ]{ \displaystyle \frac{d}{dx}\left(x^{\color{blue}{n}}\right)=\left({\color{blue}{n}}-1\right)x^{\color{blue}{n}-1} }}\) \(\color{teal}{\rm \left( with{~~~}constant{~~~}C{~~~}in{~~~}front\right)}\) \(\large{\bbox[5pt, lightyellow ,border:2px solid black ]{ \displaystyle \frac{d}{dx}\left({\rm C}\cdot x^{\color{blue}{n}}\right)={\rm C}\cdot \left({\color{blue}{n}}-1\right)x^{\color{blue}{n}-1} }}\) \(\color{blue}{\bf NOTE:{~~~~}except{~~~~}that,{~~~~}n\ne0}\) When n=0, C•x\(^0\) is a constant, and derivative of a constant is equivalent to zero. For any constant - including 0. \(------------------------------\) For example, the derivative of x\(^5\) is equivalent to (5-1)x\(^{5-1}\), and that simplifies to 4x\(^4\). AND, you would write it like this: d/dx (x\(^5\)) = 4x\(^4\) Example #2. The derivative of 3x\(^9\) is equivalent to 3(9-1)x\(^{9-1}\), and that simplifies to 3•8•x\(^{8}\), and that becomes 24x\(^8\). AND, you would write it like this: d/dx (3x\(^9\)) = 24x\(^8\)

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