Question

Use the chain rule to compute d(f ° r)/dt where f(x, y) = x² * y² and r(t) = t², t³.

Answer 1

To compute d(f ° r)/dt, you can apply the chain rule by finding the derivatives of f with respect to x and y, and the derivative of r with respect to t. Then, use the chain rule formula to compute the overall derivative and simplify the expression.

Step-by-step explanation:

To compute d(f ° r)/dt where f(x, y) = x² * y² and r(t) = t², t³,

1. Apply the chain rule to find the derivative of f with respect to x and y: ∂f/∂x = 2xy² and ∂f/∂y = 2yx².
2. Find the derivative of r with respect to t: dr/dt = 2t, 3t².
3. Use the chain rule to compute d(f ° r)/dt: d(f ° r)/dt = (∂f/∂x * dr/dt) + (∂f/∂y * dr/dt).
4. Plug in the values of ∂f/∂x, ∂f/∂y, dr/dt, and simplify the expression to obtain the final result.