Find dy/dx and d²y/dx²? x = 5 sint, y = 6 cost, 0 < t < 2π

Solution:

Given: x = 5 sint and y = 6 cost

By differentiation of parametric functions, we get dy/dx = dy/ dt . dt/dx

dx/dt = 5 cost

y = 6 cost

By differentiation

dy/dt = -6 sint

We know that

dy/dx = dy/dt/ × dx/dt

Substituting the values

dy/dx = -6 sint/ 5 cost

dy/dx = 6/5 tan t

Again by differentiation

d²y/dx² =[d/t(dy/dx)]. dt/dx

So we get,

d²y/dx² = [6/5 sec²t]/ 5 cost

= (6/25)sec³t

Therefore, dy/dx = 6/5 tan t and d²y/dx² = (6/25)sec³t

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Find dy/dx and d2y/dx2 ? x = 5 sint, y = 6 cost, 0 < t < 2π

Summary:

If x = 2 sint, y = 3 cost, 0 < t < 2π, dy/dx = 6/5 tan t and d²y/dx² = (6/25)sec³t