I find these questions about intuition to be extremely valuable. A perfect example of an intuitive explanation appears on p. 553 of Terence Tao's book Analysis II:
Intuitively, one can think of the several variable chain rule as
follows. Let $x$ be close to $x_0$. Then Newton's approximation
asserts that $$ f(x) - f(x_0) \approx f'(x_0)(x - x_0) $$ and in
particular $f(x)$ is close to $f(x_0)$. Since $g$ is differentiable at
$f(x_0)$, we see from Newton's approximation again that $$ g(f(x)) -
g(f(x_0)) \approx g'(f(x_0))(f(x) - f(x_0)). $$ Combining the two, we
obtain $$ g \circ f(x) - g \circ f(x_0) \approx g'(f(x_0)) f'(x_0)(x -
x_0) $$ which then should give $(g \circ f)'(x_0) =
g'(f(x_0))f'(x_0).$ This argument however is rather imprecise; to make
it more precise one needs to manipulate limits rigorously; see
I have often found intuitive explanations similar to this on math.stackexchange that I thought were very enlightening.
Terence Tao has also discussed three stages of mathematical education, in a blog post entitled "There's more to mathematics than rigor and proofs":
One can roughly divide mathematical education into three stages:
- The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and
hand-waving. (For instance, calculus is usually first introduced in
terms of slopes, areas, rates of change, and so forth.) The emphasis
is more on computation than on theory. This stage generally lasts
until the early undergraduate years.
- The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more
precise and formal manner (e.g. re-doing calculus by using epsilons
and deltas all over the place). The emphasis is now primarily on
theory; and one is expected to be able to comfortably manipulate
abstract mathematical objects without focusing too much on what such
objects actually “mean”. This stage usually occupies the later
undergraduate and early graduate years.
- The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready
to revisit and refine one’s pre-rigorous intuition on the subject, but
this time with the intuition solidly buttressed by rigorous theory.
(For instance, in this stage one would be able to quickly and
accurately perform computations in vector calculus by using analogies
with scalar calculus, or informal and semi-rigorous use of
infinitesimals, big-O notation, and so forth, and be able to convert
all such calculations into a rigorous argument whenever required.) The
emphasis is now on applications, intuition, and the “big picture”.
This stage usually occupies the late graduate years and beyond.
The transition from the first stage to the second is well known to be
rather traumatic, with the dreaded “proof-type questions” being the
bane of many a maths undergraduate. (See also “There’s more to maths
than grades and exams and methods".) But the transition from the
second to the third is equally important, and should not be forgotten.
Note Tao's emphasis on intuition here. I find math.stackexchange to be very helpful for developing the kind of intuition that Tao is referring to.