21.4. DIRECTIONAL AND PARTIAL DERIVATIVES 451

f (x) = 2. Therefore,

f (x+ tv)− f (x)t

=

(1+ t√

2

)2(2+ t√

2

)−2

t,

and to find the directional derivative, you take the limit of this as t → 0. However, this dif-ference quotient equals 1

4

√2(

10+4t√

2+ t2)

and so, letting t → 0,Dv f (1,2) =(

52

√2).

There is something you must keep in mind about this. The direction vector must alwaysbe a unit vector1.

21.4.2 Partial DerivativesThere are some special unit vectors which come to mind immediately. These are the vectorsei where ei = (0, · · · ,0,1,0, · · ·0)T and the 1 is in the ith position. The partial derivativesare simply directional derivatives taken in these special directions.

Definition 21.4.3 Let U be an open subset of Rn and let f : U → R. Then lettingx= (x1, · · · ,xn)

T be a typical element of Rn, ∂ f∂xi

(x)≡ Dei f (x) .This is called the partialderivative of f . Thus,

∂ f∂xi

(x) ≡ limt→0

f (x+tei)− f (x)t

= limt→0

f (x1, · · · ,xi + t, · · ·xn)− f (x1, · · · ,xi, · · ·xn)

t,

and to find the partial derivative, differentiate with respect to the variable of interest andregard all the others as constants. Other notation for this partial derivative is fxi , f,i, orDi f . If y = f (x), the partial derivative of f with respect to xi may also be denoted by ∂y

∂xior yxi or Dxi f .

Example 21.4.4 Find ∂ f∂x ,

∂ f∂y , and ∂ f

∂ z if f (x,y) = ysinx+ x2y+ z.

From the definition above, ∂ f∂x = ycosx+2xy, ∂ f

∂y = sinx+x2, and ∂ f∂ z = 1. Having taken

one partial derivative, there is no reason to stop doing it. Thus, one could take the partialderivative with respect to y of the partial derivative with respect to x, denoted by ∂ 2 f

∂y∂x or fxy.

In the above example, ∂ 2 f∂y∂x = fxy = cosx+2x. Also observe that ∂ 2 f

∂x∂y = fyx = cosx+2x.Higher order partial derivatives are defined by analogy to the above. Thus in the

above example, fyxx = −sinx+ 2. These partial derivatives, fxy are called mixed partialderivatives.

There is an interesting relationship between the directional derivatives and the partialderivatives under suitable conditions described later.

1Actually, there is a more general formulation of the notion of directional derivative known as the Gateauxderivative in which the length of v is not one but it is not considered here. This is actually a fairly old conceptsince Euler and Lagrange used something like it in their treatment of necessary conditions for the calculus ofvariations. The modern formulation is named after Gateaux who was killed in World War 1. This war killed some40 million people. When sickness and disease and famine are included, the figure is some 80 million. One outof 20 French were killed. French soldiers died at the rate of about 900 per day. It is hard to find a reason for thisconflict which would justify such an appalling loss of life.