Derivative parameric equation I would like to compute the derivative of the following parametric equations w.r.t $a$ and $b$:
$x=a~ \text{cos}(t)$ and $y= b~ \text{sin}(t)$ with $t \in [0, b]$.
Derivative w.r.t $a$ are easy to compute : $d_a x = \text{cos}(t)$, $d_a y = 0$  with $t \in [0, b]$.
However, the ones w.r.t $b$ are somewhat not intuitive since $t$ depends on $b$. 
For example, if $t=b$ then $d_b x =-ab \text{sin}(b)$
I would appreciate if someone has insight on how to compute these derivatives.
Thank you.
Reda E.

On the contrary, $t$ doesn't depend on $b.$ Or at least not for most of its values. It depends on $b$ only once in the interval $[0,1/b].$ Otherwise it doesn't. So the derivatives are as before.
