Where does the chain rule come from?

This is one of those rules that easily becomes second nature, without the need to understand what is happening at a deeper level.

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A Reminder of the Rule

Original Function

$$f(x) = (3x + 1)^5$$

Outer Function

$$g(u) = u^5$$

Inner Function

$$h(x) = 3x + 1$$

$$g'(u) = 5u^4$$

$$h'(x) = 3$$

The chain rule states: to find the derivative of nested functions, multiply their individual derivatives together.

The Chain Rule

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

The objective of this video is to explore why this is the case. Understanding the underlying concept is useful as an exercise in understanding calculus on a deeper level.

Or, if you're like me, it's fun to fantasize about whether you yourself could have come up with such a rule.

Leibniz

1676

First to formalize the chain rule

Newton

~1666

Making similar calculations a decade earlier

Building Intuition: The Gear Analogy

Imagine we have three gears: A, B, and C.

1 turn of A → 2 turns of B

1 turn of B → 3 turns of C

                    ∴ 1 turn of A → 6 turns of C
                    $$2 \times 3 = 6$$
                

Applying to Mathematics

$$f(x) = (3x + 1)^5$$

We could expand this and use the power rule term by term, but the chain rule is far quicker.

To use the gear analogy, think of everything being driven by a fundamental gear, which here is x.

The Fundamental Gear: x

X will always spin at a constant rate. One full spin represents one unit.

For the sake of this explanation, let's class a full spin of x as representing an infinitesimal — essentially a tiny, tiny change in x, which we call dx.

The Challenge

The x cog is turning the y function. But with our current setup, it's challenging to decipher how the x cog affects the rate of the function cog directly.

$$y = (3x + 1)^5$$

The Solution: A Middle Gear

The chain rule allows us to drop in a middle gear to simplify the rate finding.

Let $$u = 3x + 1$$

Then $$y = u^5$$

First Link: x → u

If the x cog spins once, the u cog spins three times because of the scaling by 3.

$$\frac{du}{dx} = 3$$

The derivative represents the instantaneous scaling of the spin rate.

Second Link: u → y

Since $y = u^5$, taking the derivative with respect to u gives:

$$\frac{dy}{du} = 5u^4$$

For example, if u = 1, the scale factor would be 5(1)⁴ = 5

That means at that moment, a single spin of u would spin y five times.

Connecting the Full Chain

For every 1 spin of x, u spins 3 times.

So we multiply our dy/du by 3 to account for all those extra spins!

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

The Final Result

Chain Rule: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Substitute: $$\frac{dy}{dx} = 5u^4 \cdot 3$$

Simplify: $$\frac{dy}{dx} = 15u^4$$

Replace u: $$\frac{dy}{dx} = 15(3x + 1)^4$$

$$\boxed{\frac{d}{dx}(3x+1)^5 = 15(3x+1)^4}$$

The Key Insight

That final multiplication isn't an arbitrary rule we just have to memorise—

it is simply the mechanical consequence of connecting these gears.

When you see that dot representing multiplication in the chain rule formula, visualise that middle gear, transmitting the spin rate from the input x all the way to the output y.

It is just the cumulative effect of rates acting upon rates.

A-LEVEL MATHS

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