Multivariable CalculusMultivariable Calculus (3).pdf

Chapter Four
Derivatives
Derivatives
Suppose f is a vector function and t0 is a point in the interior of the domain of f
( t0 in the interior of a set S of real numbers means there is an interval centered at t0 that
is a subset of S.). The derivative is defined just as it is for a plain old everyday real
valued function, except, of course, the derivative is a vector. Specifically, we say that f is
differentiable at t0 if there is a vector v such that
1
lim [ f (t0 + h) - f (t0 )] = v .
t ® t0 h
The vector v is called the derivative of f at t0 .
Now, how would we find such a thing? Suppose f (t) = a(t)i +b(t) j + c(t)k .
Then
1 æa(t + h) - a(t )ö æb(t + h) - b(t )ö æc(t + h) - c(t )ö
[ f (t + h) - f (t )] = ç 0 0 ÷i + ç 0 0 ÷j + ç 0 0 ÷k .
h 0 0 è h ø è h ø è h ø
It should now be clear that the vector function f is differentiable at t0 if and only if each
of the coordinate functions a(t),b(t), and c(t ) is. Moreover, the vector derivative v is
v = a'( t)i + b'(t) j + c'( t)k .
Now we “know” what the derivative of a vector function is, and we know how to
compute it, but what is it, really? Let’s see. Let f (t) = ti + t 3 j. This is, of course, a
vector function which describes the graph of the function y = x 3 . Let’s look at the

2
derivative of f at t0 : v = i + 3t 0 j . Convince yourself that the direction of the vector v is
3 3
the direction tangent to the graph of y = x at the point (t0 ,t0 ) . It is not so clear what
we should define to be the tangent to a curve other than a plane curve. Again, vectors
come to our rescue. If f is a vector description of a space curve, the direction of the
derivative f '( t) vector is the tangent direction at the point f (t) -the derivative f '( t) is
said to be tangent to the curve at f (t) .
If f (t) specifies the position of a particle at time t, then, of course, the derivative
is the velocity of the particle, and its length | f '(t)| is the speed. Thus the distance the
part