Problems

In Problems 1-44, find dy/dx.

45            A particle moves in the plane according to the parametric equations
x = t² + 1, y = 3t³.
Find the slope of its path.

46            An ant moves in the plane according to the equations
x = (l-t²)^-1, y = √t.
Find the slope of its path.

47            Let y depend on u, u depend on v, and v depend on x. Assume the derivatives dy/du, du/dv, and dv/dx exist. Prove that

48            Let the function f(x) be differentiable for all x, and let g(x) = f (f(x)). Show that g'(x) = f'(f(x))f'(x).