Final answer:
To find dy/dx and d2y/dx2, differentiate x and y with respect to t, express dy/dx and d2y/dx2 in terms of t, then differentiate dy/dx to find d2y/dx2.
Step-by-step explanation:
To find dy/dx and d2y/dx2 for the given parametric equations x = 5sin(t) and y = 6cos(t), we need to differentiate both equations with respect to t and then express dy/dx and d2y/dx2 in terms of t. Let's start by differentiating x and y:
- dx/dt = 5cos(t)
- dy/dt = -6sin(t)
Next, we can use the chain rule to find dy/dx:
- dy/dx = (dy/dt)/(dx/dt) = (-6sin(t))/(5cos(t)) = -6tan(t)/5
To find d2y/dx2, we differentiate dy/dx with respect to t:
- d(dy/dx)/(dt) = d(-6tan(t)/5)/(dt) = (-6sec^2(t))/5