University Maths Notes: Calculus – Partial Differentiation
Partial differentiation must be applied when we want to differentiate a function of two or more variables.
The differential of
with
respect to
is
If
instead we are given to differentiate
with
respect to
then
we treat
as
a constant, like
obtaining
The
partial derivative of
with
respect towhenis
a function of several variables is written
and the partial derivative with respect tois
written
The
partial derivatives give the gradients of the tangent lines in the
direction of the coordinate axes.
Is
the gradient of the tangent in the-
direction and
is
the gradient of the tangent in the–
direction.
Example: Find the partial derivative of
with
respect to a)and
b)
a)We differentiate using the product rule.
b)Regarding x as a constant gives
Example: Find
The derivative of
is
so
by the chain rule the derivative of
is
given by
and the derivative of
Put
to
obtain
