Derivative and integral of $u(f(t),g(t))/v(f(t),g(t))$

Question

Let $f=f(t),g=g(t) \in \mathbb{R}[t]$, let $f'=f'(t) \in \mathbb{R}[t]$ be the derivative of $f$, and let $u=u(x,y),v=v(x,y) \in \mathbb{R}[x,y]$, with $v \neq 0$.

(1) What is the derivative of $u(f(t),g(t))/v(f(t),g(t))$ with respect to $t$?

(2) What is the integral of $u(f(t),g(t))/v(f(t),g(t))$ with respect to $t$?

For example: $u(x,y)=y, v(x,y)=x, f(t)=t^2, g(t)=t^3$. Then $u(f(t),g(t))/v(f(t),g(t))=g/f=t^3/t^2=t$, so the derivative is $1$ and the integral is $\frac{t^2}{2}$.

Thank you very much!

Answer 1

Let $a(t):=u(f(t),g(t))$, $b(t):=v(f(t),g(t))$ and $h(t):=\frac{a(t)}{b(t)}$.

You are looking for $h'(t)$. This can be done with the quotient rule. Therefore you need $a'(t)$ and $b'(t)$.

For example: $a'(t)=u_x(f(t),g(t))f'(t)+u_y(f(t),g(t))g'(t)$.

Your turn !