Parametric Equations: Find $\frac{\mathrm d^2y}{\mathrm dx^2}$.

Question

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ $\displaystyle\dfrac{\mathrm d^2y}{\mathrm dx^2}=\ldots$

I don't really understand this section that I am learning at all, is there any useful website I can look over to help me understand this concept better? Thanks!

Answer 1

It is known that $$ \dfrac {\mathrm{d}y}{\mathrm{d}x} = \dfrac {\frac{\mathrm{d}y}{\mathrm{d}t}}{\frac{\mathrm{d}x}{\mathrm{d}t}}. $$Do you know how to find $y'(t)$ and $x'(t)$? Do this to get $\frac{\mathrm{d}y}{\mathrm{d}x}$ and then differentiate again. I am not finishing the whole thing for you since you haven't shown your work, but I think this hint should help you enough to finish.

Answer 2

You're given $x(t), y(t)$. So find $\dfrac{dx}{dt},\;\dfrac{dy}{dt}$.

$$\text{Then note that}\;\frac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$$

Now you need to find $\dfrac{d^2y}{dx^2}$.