DNA forms a double helix, but the curvature of this helix depends on
the temperature and on the salinity (concentration of
salt). If we wanted to understand the curvature, we would do
experiments by varying the conditions and measuring the curvature
each time. If we did our experiments well, we wouldn't try
changing both the temperature and the salinity at the same
time. Instead, we would first hold the salinity fixed and
change the temperature. Once we understood how temperature
affects curvature, we would run a second set of experiments, holding
the temperature fixed and varying the salinity. Combining the
results, we would understand how both temperature and salinity
affect curvature.

Mathematically, the curvature is a function $f(x,y)$, where $x$ is
the temperature and $y$ is the salinity. Varying the temperature
means comparing $f(x,y)$ to $f(x+h,y)$, and we can ask for the rate
of change. Varying the salinity means comparing $f(x,y)$ to
$f(x,y+h)$. By taking limits, we can compute our two partial
derivatives. We redisplay these derivatives:

Again, the definition of
partial derivatives The partial derivative of $f$ with respect to $x$ is
$\displaystyle{ \lim_{h \rightarrow 0}
\frac{f(x+h,y)-f(x,y)}{h}}.$
The partial derivative of $f$ with respect to $y$ is
$\displaystyle{ \lim_{h \rightarrow 0}
\frac{f(x,y+h)-f(x,y)}{h}}.$
In general we do not use these definitions to compute
partial derivatives.

Considering these experiments, we have applied meanings of fixing
$y$ and finding how $x$ changes ($f_x$), and then of fixing $x$ and
finding how $y$ changes ($f_y$). We don't view $f(x,y)$ as a
surface, like our previous work, but the procedure for computing the
rates of change of $x$ and of $y$ are the same.