pls. help me find the rate of change of the slope of the curve y=x^3-1 at (2,7).

Question

anonymous · Answer

Do you know how to do derivatives?

theeric · Answer

Hi! @AnImEfReaK beat me to that question.

Since the problem is a little complex, I assume this is calculus.

$\huge\sf\color{orange}{Here's\ }\large\color{green}{what\ you\ want\ to\ do}$
$\large\color{lime}{1.\ }$ You want to create an equation to give you the slope of the line. The slope is really the rate of change of the line, which you can find by derivating the original equation!
$\large\color{lime}{2.\ }$Then you want to find the rate of change of that slope from step $1$. So you derivate it again! Do it with respect to $x$ again.
$\large\color{lime}{3.\ }$You now have the equation for the rate of change of the rate of change of that original function. All you have to do now is put your point's $x$ value in (because you've been derivating with respect to $x$), and see what you get from that function! 

:)

theeric · Answer

I think that is how you find your solution, at least :)

anonymous · Answer

@theEric I didn't know you had to find the derivative twice...

theeric · Answer

Hi! Thanks for checking that out! :)

I still think you do have to do two derivatives.

You have a function, $y=x^3-1$, and you want to find the rate of change of the slope! The slope that is tangent to the function, I mean that instantaneous slope at one point, is the rate of change of $y=x^3-1$, right? That means you can derivate $y=x^3-1$ to get the slope.

Then, to get the rate of change of the slope, you need to derivate that, too. So $\dfrac{dy}{dx}=3x^2$, right? And that's the slope? Then you derivate that to get $\it\color{blue}{its}$ rate of change. That's what the second derivative is for.

Then, after both of those, you have the rate of change of the rate of change of the function, a.k.a. rate of change of the slope. You can then insert your $x$ since that's what we derivated with respect to, and find your answer as the $\dfrac{ddy}{dxx}=\left(\text{derivative of slope}\right)$.

You know what I mean? Is there still any discrepancy? Thanks for replying! :)