Given the parametric equations x = lnt+t and y = sint calculate d^2y/dx^2
First we can write d2y/dx2 as (d/dx)(dy/dx). Now we need to find dy/dx. This can be further written as (dy/dt)(dt/dx). These derivatives can be obtained from the given parametric equations:
dx/dt = 1/t + 1, hence dt/dx = 1/(1/t+1) = t/(1+t)
dy/dt = cost,
Therefore combining these to we obtain
dy/dx = tcost/(1+t).
Now goind back to what we wrote at the beginning - d2y/dx2 as (d/dx)(dy/dx) - we can write it as (dt/dx)(d/dx)(dy/dx) and this is equal to (dt/dx)(d/dt)[tcost/(t+1)].
First let's compute (d/dt)[tcost/(t+1)]:
Using the product and quotient rule (if we have to functions say f and g and their derivatives f' and g' product rule says that (fg)' = f'g + fg' and quotient rule says that (f/g) = (f'g-fg')/g2) we can write:
[tcost/(t+1)]' = (cost-tsint)(t+1)-tcost)/(t+1)2 = (cost-t2sint-tsint)/(t+1)2
Going bakc to the formula for the second derivative we need to multiply our result by dt/dx, therefore
(cost-t2sint-tsint)/(t+1)2 (dt/dx) = (cost-t2sint-tsint)/(t+1)2 * [t/(1+t)] = t(cost-t2sint-tsint)/(t+1)3
Final answer: d2y/dx2 = t(cost-t2sint-tsint)/(t+1)3
