Differentiable
For a single-variable function , if is differentiable at , then the
graph of will have a tangent line at . The tangent line’s equation is:
The same idea can be generalized to two-variable functions. They will have a
tangent plane instead of a line.
Suppose . If and exists and can
be expressed in the below form, then is differentiable at .
where and approach as
approach .
is said to be differentiable if it is differentiable at every point in its
domain.
If partial derivatives of is not existent at a point, then will
not be differentiable at that point.
If and are continuous throughout at an open region , then
is differentiable at every point of .
The differentiability can also be proven by proving:
where .
Implies Continuity
Equation of tangent plane
For the function at .
Derivative
Suppose where is a differentiable function of and .
is called the total differential of .
Chain Rule
Let be single variable differentiable functions and be a 2
variable differentiable function having continuous first order partial
derivatives.
is differentiable.
Can be extended for functions of more than 2 variables.