1. ## parametric derivative

If x=t^3, y=2t^2 for t>0, then what does d^2y/dx^2 equal?

2. $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$
$\frac{d^2y}{dx^2} = \frac{d}{dt} \left ( \frac{dy}{dx} \right ) \cdot \frac{dt}{dx}$

3. This is what the question is asking.
$\frac{{d^2 y}}{{dx^2 }} = \frac{d}{{dx}}\left( {\frac{{dy}}{{dx}}} \right) = \frac{{\frac{d}{{dt}}\left( {\frac{{dy}}{{dx}}} \right)}}{{\frac{{dx}}{{dt}}}}$
Step 1. Express $y' = \frac{{dy}}{{dx}}$ in terms of t.

Step 2. find $\frac{{dy'}}{{dt}}$, i.e. the derivative of step 1 with respect to t.

Step 3. divide $\frac{{dy'}}{{dt}}$ by $\frac{{dx}}{{dt}}$.

4. So is this what I should do?
$y=\frac{{2x}}{{t}}
$

Step 1: $
y'=\frac{{2}}{{t}}
$

Step 2: $
\frac{{dy'}}{{dt}}=\frac{{-2}}{{t^2}}
$

Step 3: $
\frac{{dx}}{{dt}}=3t^2
$

so the answer would be $\frac{{d^2y}}{{dx^2}}=\frac{{-2}}{{3t^4}}
$
.
Is that right?

So is this what I should do?
$y=\frac{{2x}}{{t}}
$

Step 1: $
y'=\frac{{2}}{{t}}
$
No, this is incorrect, you cannot treat t as a constant when you know it is a variable
Step 2: $
\frac{{dy'}}{{dt}}=\frac{{-2}}{{t^2}}
$

Step 3: $
\frac{{dx}}{{dt}}=3t^2
$

so the answer would be $\frac{{d^2y}}{{dx^2}}=\frac{{-2}}{{3t^4}}
$
.
Is that right?

first find $\frac{dx}{dt}$ and $\frac{dy}{dt}$

you should get $\frac{dx}{dt} = 3t^2$ and $\frac{dy}{dt} = 4t$

then you use $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$

giving $\frac{dy}{dx} = 4t \cdot \frac{1}{3t^2}$

want to finish this off ?

6. Okay, so $
\frac{dy}{dx} = \frac{4}{3t}
$

Step 2: $
\frac{{dy'}}{{dt}}=\frac{{-4}}{{3t^2}}
$

Step 3: $
\frac{{dx}}{{dt}}=3t^2
$

so the answer would be $
\frac{{d^2y}}{{dx^2}}=\frac{{-4}}{{9t^4}}
$
.
Is that right?

7. Yeah that looks good to me.

Okay, so $
\frac{dy}{dx} = \frac{4}{3t}
$

Step 2: $
\frac{{dy'}}{{dt}}=\frac{{-4}}{{3t^2}}
$

Step 3: $
\frac{{dx}}{{dt}}=3t^2
$

so the answer would be $
\frac{{d^2y}}{{dx^2}}=\frac{{-4}}{{9t^4}}
$
.
Is that right?
By the way .... since y = 2t^2 you can express the result as a function of y ......