Answer
$$
\frac{\partial R}{\partial L}=C\frac{1}{r^{4}}.
$$
and
$$
\begin{aligned}
\frac{\partial R}{\partial r}=\frac{-4 C L}{r^{5}}
\end{aligned}
$$
Work Step by Step
One of Poiseuille’s laws states that the resistance of blood fowing through an artery is
$$
R=C\frac{L}{r^{4}}, \quad\quad\quad\quad (1)
$$
where $L$ and $r$ are the length and radius of the artery and $C$ is a positive constant determined by the viscosity of the blood.
To find $\frac{\partial R}{\partial L}$, we differentiate explicitly with respect to $L$, being careful to treat $r$ as a constant:
$$
\frac{\partial R}{\partial L}=C\frac{1}{r^{4}}.
$$
Now, we rewrite Eq. (1) as follows
$$
R=C L r^{-4}
$$
To find $\frac{\partial R}{\partial r}$, we differentiate explicitly with respect to $r$, being careful to treat $L$ as a constant:
$$
\begin{aligned}
\frac{\partial R}{\partial r}&=-4 C L (r)^{-5} \\
&=\frac{-4 C L}{r^{5}}
\end{aligned}
$$
