Tangent Plane

The partial derivative with respect to x at a point in R3 measures the rate of change of the
function along the X-axis or say along the direction (1; 0; 0). We will now see that this notion can
be generalized to any direction in R3.
Directional Derivative : Let f : R3 ! R; X0 2 R3 and U 2 R3 such that k U k = 1. The
directional derivative of f in the direction U at X0 = (x0; y0; z0) is de¯ned by
DX0f(U) = lim
t!0
f(X0 + tU) ? f(X0)
t
provided the limit exists.
It is clear that DX0f(e1) = fx(X0); DX0f(e2) = fy(X0) and DX0f(e3) = fz(X0):
The proof of the following theorem is similar to the proof of Theorem 26.2.
Theorem 28.1: If f is di®erentiable at X0, then DX0f(U) exists for all U 2 R3; k U k= 1:
Moreover, DX0f(U) = f0(X0) ¢ U = (fx(X0); fy(X0); fz(X0)) ¢ U:
The previous theorem says that if a function is di®erentiable then all its directional derivatives
exist and they can be easily computed from the derivative.
(i) In this example we will see that a function is not di®erentiable at a point but the directional
derivatives in all directions at that point exist.
De¯ne f : R2 ! R by f(x; y) = x2y
x4+y2 when (x; y) 6= (0; 0) and f(0; 0) = 0.
This function is not continuous at (0; 0) and hence it is not di®erentiable at (0; 0).
We will show that the directional derivatives in all directions at (0; 0) exist. Let U = (u1; u2) 2
R3; k U k = 1 and 0 = (0; 0). Then
lim
t!0
f(0 + tU) ? f(0)
t
= lim
t!0
t3u2
1 u2
t(t4u4
1 + t2u2
2 )
= lim
t!0
u2
1u2
t2u4
1 + u2
2
= 0; if u2 = 0 and u2
1
u2
; if u2 6= 0
Therefore, D0f((u1; 0)) = 0 and D0f((u1; u2)) = u2
1
u2
when u2 6= 0.
(ii) In this example we will see that the directional derivative at a point with respect to some vector
may exist and with respect to some other vector may not exist.