(iii) Let $\tilde\gamma(\tilde t)$ be a reparametrization of $\gamma(t)$, and suppose $\gamma$ has an ordinary cusp at $t=t_0$. Then at $t=t_0$, $$d\tilde\gamma/d\tilde t=(d\gamma/dt)(dt/d\tilde t)=0,\quad d^2\tilde\gamma/d\tilde t^2=(d^2\gamma/dt^2)(dt/d\tilde t)^2,$$ $$d^3\tilde\gamma/d\tilde t^3=(d^3\gamma/dt^3)(dt/d\tilde t)^3+3(d^2\gamma/dt^2)(dt/d\tilde t)(d^2t/d\tilde t^2).$$ Using the fact that $dt/d\tilde t\ne0$, it is easy to see that $d^2\tilde\gamma/d\tilde t^2$ and $d^3\tilde\gamma/d\tilde t^3$ are linearly independent when $t=t_0$.
I understand that if a curve has a regular cusp at a point then it’s first derivative =0 however i don’t understand the 2nd and 3rd derivatives for 2 reasons
1)there are missing terms if I do the standard chain rule with equivalent notation, for example on the 2nd derivative. $$g'(x)^2f''(g(x))+g''(x)f'(g(x))$$
The 3rd derivative is also missing terms.
2)how does one see that the 2nd and 3rd derivatives are linearly independent?