(a) This follows from the definition for the partial derivative, with the help of some limit properties and a well-known limit.
• Recall that for  , we have the partial derivative with respect to
, we have the partial derivative with respect to  defined as
 defined as

The derivative at (0, 0) is then

• By definition of  ,
,  , so
, so

• Expanding the tangent in terms of sine and cosine gives

• Introduce a factor of  in the numerator, then distribute the limit over the resulting product as
 in the numerator, then distribute the limit over the resulting product as

• The first limit is 1; recall that for  , we have
, we have

The second limit is also 1, which should be obvious.
• In the remaining limit, we end up with

and this is exactly the partial derivative of  with respect to
 with respect to  .
.

For the same reasons shown above,

(b) To show that  is differentiable at (0, 0), we first need to show that
 is differentiable at (0, 0), we first need to show that  is continuous.
 is continuous.
• By definition of continuity, we need to show that

is very small, and that as we move the point  closer to the origin,
 closer to the origin,  converges to
 converges to  .
.
We have

The first expression in the product is bounded above by 1, since  for all
 for all  . Then as
. Then as  approaches the origin,
 approaches the origin,

So,  is continuous at the origin.
 is continuous at the origin.
• Now that we have continuity established, we need to show that the derivative exists at (0, 0), which amounts to showing that the rate at which  changes as we move the point
 changes as we move the point  closer to the origin, given by
 closer to the origin, given by

approaches 0.
Just like before,

and this converges to  , since differentiability of
, since differentiability of  means
 means

So,  is differentiable at (0, 0).
 is differentiable at (0, 0).