21-259: Calculus in Three Dimensions
Lecture #9
Spring 2025
The Chain Rule
Recall that when y = f (x) and x = g (t) where f and g are differentiable functions, that y is also a differentiable function of t as y = f (g (t)) and thus the Chain Rule gives
The same is true for multivariate functions, however there may be more than one “path” to the inde-pendent variable(s).
The Chain Rule (Case I): Suppose that z = f (x, y) is a differen-tiable function of x and y, where x = g (t) and y = h(t) are both differentiable functions of t. Then z is a differentiable function of t and
Example 1. Find if z = arctan(y/x), x = e
t
, and y = 1−e
−t
.
Example 2. Find when t = 1 if w = xe y/z
, x = t
2
, y = 1− t, and z = 1+2t.
The Chain Rule (Case II): Suppose that z = f (x, y) is a differentiable function of x and y, where x = g (s,t) and y = h(s,t) are both differentiable functions of s and t. Then z is a differentiable function of s and t, and
Example 3. Let z = x
2+x y3 where x = w v2+w and y = u +vew . Find and when u = 2, v = 1,w = 0.
If z = f (x, y) is an implicitly-defined function by an equation of the form. F(x, y, z) = 0, then we can use the chain rule to compute the derivative with respect to x or y implicitly:
Since ∂x/∂x = 1 and ∂y/∂x = 0, we have which can be solved for .
Theorem: If F(x, y, z) = 0 defines z implicitly as a function of x and y, and the below partial derivatives all exist, then
This can also by used to find when
Example 4. Find
Example 5. Find
Directional Derivatives and the Gradient Vector
Definition: The directional derivative of f (x, y) at (x0, y0) in the direction of a unit vector u =〈a,b〉 is
If f = f (x, y, z), the directional derivative in the direction of the unit vector u = 〈a,b,c〉, then
In general, the directional derivative of f at x0 in the direction of the unit vector u is
Theorem: If f is a differentiable function of x and y, then f has directional derivative in the direction of unit vector u = 〈a,b〉 and
Du f (x, y) = fx (x, y)a + f y (x, y)b.
If f is a differentiable function of x, y and z, then f has directional derivative in the direction of unit vector u = 〈a,b,c〉 and
Du f (x, y, z) = fx (x, y, z)a + f y (x, y, z)b + fz (x, y, z)c.
Example 6. Find the directional derivative of the function f (x, y) = x
2
y
3 −4y at the point (2,−1) in the direction of the vector v = 2ı +5ȷ.
Example 7. Find the directional derivative of the function f (x, y) = x
3 − 3x y + 4y
2
in the direction of the unit vector that makes an angle of π/6 with the positive x-axis.
Example 8. Find Du f at the point (1,3,1) if
Definition: Let f be a differentiable function of multiple variables x1,...,xn. The gradient of f is the vector function ∇f given by
For example, if f = f (x, y), ∇f =
〈fx , f y〉, and if f = f (x, y, z), then ∇f = 〈fx , f y , fz〉. An alternate notation for ∇f is gradf .
The operator ∇ (pronounced “grad” or “del”) is a differential operator, i.e., it is something that we can apply to functions to produce other functions. Specifically, in three dimensions,
so ∇f is really right scalar multiplication of ∇ by f . Note: we have that Du f = ∇f ·u.
Example 9. If f (x, y, z) = x sin yz, find ∇f and the directional derivative of f at (1,3,0) in the direction of v = ı +2ȷ −k.
Theorem: Let f be a differentiable function of multiple variables. The maximum value of Du f (x) is |∇f (x)| and it occurs when u has the same direction as ∇f (x).
Example 10. Suppose that the temperature at a point (x, y, z) in space is given by
T (x, y, z) = 80/(1+ x
2 +2y
2 +3z
2
)
2
,
where T is measured in degrees Celsius and x, y, z are in meters. In which direction does the temper-ature increase the fastest at the point (1,1,−2)? What is the maximum rate of increase?
The gradient gives the direction of maximum increase of a function at a point. Evaluated at x0, the vector ∇f (x0) is orthogonal to the level curves/surfaces of f that pass through the point x0.
Example 11. Find the equations of the tangent plane and the nor-mal line at the point (−2,1,−3) to the ellipsoid