If x=7t^2 - 9t , y=t^6+2t^2 then dy/dx=... and d^2y/dx^2=...?

x = 7t² - 9t
dx/dt = 14t - 9

y = t⁶ + 2t²
dy/dt = 6t⁵ + 4t

dy/dx
= (dy/dt) / (dx/dt)
= (6t⁵ + 4t) / (14t - 9)

d²y/dx
= (d/dx)[(6t⁵ + 4t) / (14t - 9)]
= [(14t - 9)(6t⁵ + 4t)' - (6t⁵ + 4t)(14t - 9)'] / (14t - 9)²
= [(14t - 9)(30t⁴ + 4) - (6t⁵ + 4t)(14)] / (14t - 9)²
= [(420t⁵ - 270t⁴ + 56t - 36) - (84t⁵ + 56)] / (14t - 9)²
= (336t⁵ - 270t⁴ + 56t - 92) / (14t - 9)²
= 2(168t⁵ - 135t⁴ + 28t - 48) / (14t - 9)²

x = 7t² - 9t and y = t⁶ + 2t²

so, dx/dt = 14t - 9 and dy/dt = 6t⁵ + 4t

Hence, dy/dx = (6t⁵ + 4t)/(14t - 9)

Now, (d/dx)(dy/dx) => (d/dt)(dy/dx)(dt/dx) 

Hence, (d/dt)[(6t⁵ + 4t)/(14t - 9)](dt/dx)

or, (d/dt)[(6t⁵ + 4t)/(14t - 9)]/(dx/dt)

First, we do (d/dt)[(6t⁵ + 4t)/(14t - 9)] using the 'quotient rule'

so, [(30t⁴ + 4)(14t - 9) - (14)(6t⁵ + 4t)]/(14t - 9)²

Then, dividing by dx/dt => (14t - 9) we get:

[(30t⁴ + 4)(14t - 9) - (14)(6t⁵ + 4t)]/(14t - 9)³

Tidying up the numerator we get:

[420t⁵ - 270t⁴ + 56t - 36 - 84t⁵ - 56t]

i.e. 336t⁵ - 270t⁴ - 36

Then, d²y/dx² = (336t⁵ - 270t⁴ - 36)/(14t - 9)³

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