## anonymous one year ago 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!

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1. anonymous

Yes thats the method i was using

2. anonymous

i cant get it though

3. Hero

@jobo What's f(x+h) ?

4. anonymous

I got $x ^{3}-3x ^{2}\Delta x + 3x \Delta x ^{2} - \Delta x ^{3} -x-\Delta x -x ^{3} +x$

5. 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.

6. IrishBoy123

and the answer is not $$\text{x^2-1}$$ :p

7. 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$$^8$$