Question

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!

Answer 1

Yes
thats the method i was using

Answer 2

i cant get it though

Answer 3

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

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

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\(^8\)