Originally Posted by
Opalg I meant differentiate with respect to x, using the chain rule. I suppose I should have used t rather than x, since the question says that f is a function of t. Then $\displaystyle \tfrac d{dt}(f_1(t))^2$ evaluated at t=a is equal to $\displaystyle 2f_1(a)\tfrac {df_1}{dt}(a)$, and similarly for the other coordinates. The result of differentiating both sides of the equation $\displaystyle (f_1(t))^2 + (f_2(t))^2 + \ldots + (f_n(t))^2 = r^2$ in that way ought to remind you of an inner product.