The derivative of the function at
is the limit of the difference quotient as
approaches zero.
If the limit exists, then
is differentiable at
.
In calculus, the differential represents the principal part of the change in a function with respect to changes
in the independent variable. The differential
is defined by:
where is the derivative of
with respect to
, and
is an additional real variable
(so that
is a function of
and
). The notation is such that the equation
holds, where the derivative is represented in the Leibniz notation , and this is consistent with
regarding the derivative as the quotient of the differentials. One also writes
The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical
rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a
particular differential form, or analytical significance if the differential is regarded as a linear approximation to the
increment of a function. In physical applications, the variables and
are often constrained to be
very small (“infinitesimal”).
- Lagrange’s notation:
- Leibnitz’s notation:
The sum(or difference) of two differentiable functions is differentiable and is the sum(or difference) of their derivatives.
If is a differentiable function and
is a real number, then
is also differentiable.
The product of two differentiable functions, and
, is itself differentiable. Moreover, the derivative
of
is the first function times the derivative of the second, plus the second function times the derivative of
the first.
The quotient , of two differentiable functions,
and
, is itself differentiable at
all values of
for which
. Moreover, the derivative of
is given by the
denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by
the square of the denominator.
If is a differentiable function of
and
is a differentiable function of
,
then
is a differentiable function of
and