Differentiation of Quotients and Higher Derivatives

[Aichuk]

1) The line 2x+9=3 meets curve xy+y+2=0 at the points P and Q. Calculate the gradient of the curve at P and Q

2)Given that $y=(x^2)/(x-2)$, find
a) $(d^2)y/dx^2$ in its simplest form
b)ther range of value for which $dy/dx$ and $(d^2)y/dx^2$ are positive.

I can't figure out either of the sums. For the first one I got the answer -1/11 & -44/81 (even though the answer page showed 1/2 and 4/81) and the second I couldn't do. Can anyone do them step by step?

 

[ehild]

What is the first derivative of $y=\frac{x^2}{x-2}$

ehild

 

[Aichuk]

y=(x^2)-4x/(x-2)^2

 

[Ray Vickson]

y=(x^2)-4x/(x-2)^2

How did you get
$$\frac{d}{dx} \frac{x^2}{x-2} = x^2 - \frac{4x}{(x-1)^2} ?$$
This is obviously wrong.

 

[Aichuk]

No, i got $y= \frac{x^2-4x}{(x-2)^2}$

 

[HallsofIvy]

1) The line 2x+9=3 meets curve xy+y+2=0 at the points P and Q. Calculate the gradient of the curve at P and Q

Do you mean "2x+ 9y= 3"? What are P and Q?

2)Given that $y=(x^2)/(x-2)$, find
a) $(d^2)y/dx^2$ in its simplest form
b)ther range of value for which $dy/dx$ and $(d^2)y/dx^2$ are positive.

I can't figure out either of the sums. For the first one I got the answer -1/11 & -44/81 (even though the answer page showed 1/2 and 4/81) and the second I couldn't do. Can anyone do them step by step?

As Ray Vickson pointed out, you have written the first derivative incorrectly- although you may have calculated it correctly. You have the parentheses in the wrong place and you have the denominator wrong.

 

[Aichuk]

Sorry for that, I suck at typing equations on computer.

 

[Mark44]

Sorry for that, I suck at typing equations on computer.

Here's how to get better at it - https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[ehild]

No, i got $y= \frac{x^2-4x}{(x-2)^2}$

Did you mean $\frac{dy}{dx}$?ehild