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
