Chain Rule

Statement

The Chain Rule is a theorem of calculus which states that if $h(x) = f(g(x))$ , then $h'(x)=f'(g(x))\cdot g'(x)$ wherever those expressions make sense.

For example, if $f(x)=\sin{x}$ , $g(x)=x^2$ , and $h(x)=f(g(x))=\sin{(x^2)}$ , then $h'(x) = \cos{(x^2)}\cdot(2x)$ .

Here are some more precise statements for the single-variable and multi-variable cases.

Single variable Chain Rule:

Let each of $I \subset \mathbb{R}, J \subset \mathbb{R}$ be an open interval, and suppose $g:I \to J$ and $f:J \to \mathbb{R}$ . Let $h:I \to \mathbb{R}$ such that $h(x) = f(g(x)) \forall x \in I$ . If $x_0 \in I$ , $g$ is differentiable at ${x_0}$ , and ${f}$ is differentiable at $g(x_0),$ then ${h}$ is differentiable at ${x_0}$ , and ${h'(x_0) = f'(g(x_0))\cdot g'(x_0)}$ .

Multi-dimensional Chain Rule:

Let $g:\mathbb{R}^n \to \mathbb{R}^m$ and $f:\mathbb{R}^m \to \mathbb{R}^p$ . (Here each of $n$ , $m$ , and ${p}$ is a positive integer.) Let ${h}: \mathbb{R}^n \to \mathbb{R}^p$ such that $h(x) = f(g(x)) \forall x \in \mathbb{R}^n$ . Let $x_0 \in \mathbb{R}^n$ . If $g$ is differentiable at ${x_0}$ , and ${f}$ is differentiable at $g(x_0),$ then $h$ is differentiable at ${x_0}$ and $h'(x_0) = f'(g(x_0))\cdot g'(x_0)$ . (Here, each of $h'(x_0)$ , $f'(g(x_0))$ , and $g'(x_0)$ is a matrix.)

Intuitive Explanation

The single-variable Chain Rule is often explained by pointing out that

$\frac{f(g(x+\Delta x)) - f(g(x))}{\Delta x} = \frac{f(g(x+\Delta x)) - f(g(x))}{g(x+ \Delta x)-g(x)}\cdot \frac{g(x+ \Delta x)-g(x)}{\Delta x}$ .

The first term on the right approaches $f'(g(x))$ , and the second term on the right approaches $g'(x)$ , as $\Delta x$ approaches $0$ . This can be made into a rigorous proof. (But we do have to worry about the possibility that $g(x+\Delta x) - g(x)=0$ , in which case we would be dividing by $0$ .)

This explanation of the chain rule fails in the multi-dimensional case, because in the multi-dimensional case $\Delta x$ is a vector, as is $g(x+\Delta x) - g(x)$ , and we can't divide by a vector.

However, there's another way to look at it.

Suppose a function $F$ is differentiable at $x$ , and $\Delta x$ is "small". Question: How much does $F$ change when its input changes from $x$ to $x+ \Delta x$ ? (In other words, what is $F(x+ \Delta x) - F(x)$ ?) Answer: approximately $F'(x) \cdot \Delta x$ . This is true in the multi-dimensional case as well as in the single-variable case.

Well, suppose that (as above) $h(x) = f(g(x))$ , and $\Delta x$ is "small", and someone asks you how much $h$ changes when its input changes from $x$ to $x+ \Delta x$ . That is the same as asking how much $f$ changes when its input changes from $g(x)$ to $g(x+ \Delta x)$ , which is the same as asking how much $f$ changes when its input changes from $g(x)$ to $g(x) + \Delta g$ , where $\Delta g = g(x+ \Delta x) - g(x)$ . And what is the answer to this question? The answer is: approximately, $f'(g(x)) \cdot \Delta g$ .

But what is $\Delta g$ ? In other words, how much does $g$ change when its input changes from $x$ to $x+ \Delta x$ ? Answer: approximately $g'(x) \cdot \Delta x$ .

Therefore, the amount that $h$ changes when its input changes from $x$ to $x+ \Delta x$ is approximately ${f'(g(x)) \cdot g'(x) \cdot \Delta x}$ .

We know that $h'(x)$ is supposed to be a matrix (or number, in the single-variable case) such that $h'(x) \cdot \Delta x$ is a good approximation to $h(x+ \Delta x) - h(x)$ . Thus, it seems that $f'(g(x)) \cdot g'(x)$ is a good candidate for being the matrix (or number) that $h'(x)$ is supposed to be.

This can be made into a rigorous proof. The standard proof of the multi-dimensional chain rule can be thought of in this way.

Proof

Here's a proof of the multi-variable Chain Rule. It's kind of a "rigorized" version of the intuitive argument given above.

I'll use the following fact. Assume $F: \mathbb{R}^n \to \mathbb{R}^m$ , and $x \in \mathbb{R}^n$ . Then $F$ is differentiable at ${x}$ if and only if there exists an $m$ by $n$ matrix $M$ such that the "error" function ${E_F(\Delta x)= F(x+\Delta x)-F(x)-M\cdot \Delta x}$ has the property that $\frac{|E_F(\Delta x)|}{|\Delta x|}$ approaches $0$ as $\Delta x$ approaches $0$ . (In fact, this can be taken as a definition of the statement " $F$ is differentiable at ${x}$ .") If such a matrix $M$ exists, then it is unique, and it is called $F'(x)$ . Intuitively, the fact that $\frac{|E_F(\Delta x)|}{|\Delta x|}$ approaches $0$ as $\Delta x$ approaches $0$ just means that $F(x + \Delta x)-F(x)$ is approximated well by $M \cdot \Delta x$ .

Okay, here's the proof.

Let $g:\mathbb{R}^n \to \mathbb{R}^m$ and $f:\mathbb{R}^m \to \mathbb{R}^p$ . (Here each of $n$ , $m$ , and ${p}$ is a positive integer.) Let ${h}: \mathbb{R}^n \to \mathbb{R}^p$ such that $h(x) = f(g(x)) \forall x \in \mathbb{R}^n$ . Let $x_0 \in \mathbb{R}^n$ , and suppose that $g$ is differentiable at ${x_0}$ and $f$ is differentiable at $g(x_0)$ .

In the intuitive argument, we said that if $\Delta x$ is "small", then $\Delta h = f(g(x_0+\Delta x))-f(g(x_0)) \approx f'(g(x_0))\cdot \Delta g$ , where $\Delta g = g(x_0+\Delta x)-g(x_0)$ . In this proof, we'll fix that statement up and make it rigorous. What we can say is, if $\Delta x \in \mathbb{R}^n$ , then $\Delta h = f(g(x_0)+\Delta g)-f(g(x_0)) = f'(g(x_0))\cdot \Delta g + E_f(\Delta g)$ , where $E_f:\mathbb{R}^m \to \mathbb{R}^p$ is a function which has the property that $\lim_{\Delta g \to 0} \frac{|E_f(\Delta g)|}{|\Delta g|}=0$ .

Now let's work on $\Delta g$ . In the intuitive argument, we said that $\Delta g \approx g'(x_0)\cdot \Delta x$ . In this proof, we'll make that rigorous by saying $\Delta g = g'(x_0)\cdot \Delta x + E_g(\Delta x)$ , where $E_g:\mathbb{R}^n \to \mathbb{R}^m$ has the property that $\lim_{\Delta x \to 0} \frac{|E_g(\Delta x)|}{\Delta x} = 0$ .

Putting these pieces together, we find that $\Delta h = f'(g(x_0))\Delta g + E_f(\Delta g)$ $= f'(g(x_0))\left(g'(x_0)\Delta x + E_g(\Delta x)\right) + E_f \left( g'(x_0)\Delta x + E_g(\Delta x) \right)$ $=f'(g(x_0))g'(x_0)\Delta x + f'(g(x_0))E_g(\Delta x) + E_f \left( g'(x_0)\Delta x + E_g(\Delta x) \right)$ $= f'(g(x_0))g'(x_0)\Delta x + E_h(\Delta x)$ , where I have taken that messy error term and called it $E_h(\Delta x)$ .

Now, we just need to show that $\frac{|E_h(\Delta x)|}{|\Delta x|} \to 0$ as $\Delta x \to 0$ , in order to prove that $h$ is differentiable at ${x_0}$ and that $h'(x_0) = f'(g(x_0))g'(x_0)$ .

I believe we've hit a point where intuition no longer guides us. In order to finish off the proof, we just need to look at $E_h(\Delta x)$ and play around with it a bit. It's not that bad. For the time being, I'll leave the rest of the proof as an exercise for the reader. (Hint: If $A$ is an $m$ by $n$ matrix, then there exists a number $k > 0$ such that $|Ax| \le k|x|$ for all $x \in \mathbb{R}^n$ .)

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Chain Rule

Contents

Statement

Intuitive Explanation

Proof

See also