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. For functions with more than 2 variables, they
have a tangent space.
Differentiable
Suppose where ;
and .
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 iff it is differentiable at every point in
its domain.
If either or is non-existent at a point, then is not
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:
Implies Continuity
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.