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The Multivariable Chain Rule
Nikhil Srivastava
February 11, 2015
The chain rule is a simple consequence of the fact that differentiation produces the linear approximation to a function at a point, and that the derivative is the coefficient appearing in this linear approximation. Let’s see this for the single variable case first. It is especially transparent using o() notation, where once again f (x) = o(g(x)) means that
lim x→ 0
f (x) g(x)
Suppose we are interested in computing the derivative of (f ◦ g)(x) = f (g(x)) at x, where f and g are both differentiable functions from R to R. Since g is differentiable, we have (by the definition of differentiation as a limit):
g(x + ∆x) = g(x) + g′(x)∆x + o(∆x)
for a number g′(x) which we call the derivative of g at x. In words, this says that g is well-approximated by its linear approximation in a neighborhood of x. Similarly, we have
f (y + ∆y) = f (y) + f ′(y)∆y + o(∆y).
Letting y = g(x) and ∆y = g′(x)∆x + o(∆x), we now find that
(f ◦ g)(x + ∆x) = f (g(x + ∆x))
= f (g(x) + g′(x)∆x + o(∆x)) = f (g(x)) + f ′(g(x)) (g′(x)∆x + o(∆x)) + o(∆y) = f (g(x)) + f ′(g(x)) · g′(x)∆x + o(∆x) + o(∆y) since f ′(g(x)) · o(∆x) = o(∆x).
Thus, we have
lim ∆x→ 0
(f ◦ g)(x + ∆x) − (f ◦ g)(x) ∆x
= lim ∆x→ 0
f ′(g(x)) · g′(x) + o(∆x) + o(∆y) ∆x
= f ′(g(x))·g′(x),
establishing the chain rule.
A very similar thing happens in the multivariable case. Suppose f : R^2 → R and g : R^2 → R^2 are differentiable. To parallel the notation used in class, let z = f (x, y) and (x, y) = g(s, t). Since both functions are differentiable, they must have linear approximations:
f (x + ∆x, y + ∆y) = f ((x, y) + (∆x, ∆y)) ≈ f (x, y) + Lf (∆x, ∆y) (∗), g(s + ∆s, t + ∆t) = g((s, t) + (∆s, ∆t)) ≈ g(s, t) + Lg(∆s, ∆t) (∗∗)
where Lf : R^2 → R and LG : R^2 → R^2 are linear functions, and I have used ≈ to indicate equality up to o(∆x) terms^1. But we know that all linear functions are implemented by matrices so there must be a 1 × 2 matrix Df such that
Lf (∆x, ∆y) = Df
[
∆x ∆y
]
In fact, we know exactly what this matrix is (by comparing coefficients):
Df =
[
∂f ∂x
∂f ∂y
]
so that we have the explicit formula
Lf (∆x, ∆y) =
[
∂f ∂x
∂f ∂y
] [∆x ∆y
]
∂f ∂x
∆x +
∂f ∂y
∆y,
which is the same as what is given by the total differential df. Repeating this process for Lg, we get that for the 2 × 2 matrix
Dg =
[∂x ∂s
∂x ∂y ∂t ∂s
∂y ∂t
]
the linear approximation of g at (s, t) is given by
Lg(∆s, ∆t) = Dg
[
∆s ∆t
]
[∂x ∂s
∂x ∂y ∂t ∂s
∂y ∂t
] [
∆s ∆t
]
[∂x ∂s ∆s^ +^
∂x ∂y ∂t^ ∆t ∂s ∆s^ +^
∂y ∂t ∆t
]
Now for the punch line: just as in the univariate case, we write:
(f ◦ g)((s, t) + (∆s, ∆t)) = f (g((s, t) + (∆s, ∆t)))
≈ f
g(s, t) + Dg
[
∆s ∆t
])
by (∗∗)
≈ f (g(s, t)) + Df Dg
[
∆s ∆t
]
by (∗), treating Dg
[
∆s ∆t
]
as
[
∆x ∆y
]
(^1) There is a subtlety about uniform convergence vs pointwise convergence here, but for the purposes of
this course you can ignore it, and what is written here is good enough.
