I have so many thoughts about teaching the chain rule. If you just want some problems to help you introduce the chain rule in a problem-based way, I can cut to the chase: start with these problems and then move on to these ones. If you look at those problems and they make sense, you can probably use them without reading more. But if you want some extensive rumination on teaching the chain rule, I have a whole series of posts coming up for you.
To my mind, half the battle of teaching the chain rule is helping students understand how it is that the Leibniz form and the prime form mean the same thing. On the one hand, $$\frac{{\rm d}y}{{\rm d}x}=\frac{{\rm d}y}{{\rm d}u}\cdot\frac{{\rm d}u}{{\rm d}x}$$ feels so obvious that students can’t quite see why one would bother commenting on it. But this is deceptive: derivatives aren’t just fractions and the importance of the fact that you can treat them like they are is deep and vast. On the other hand, $$(f(u))'(t)=f'(u(t))\cdot u'(t)$$ feels impenetrable: it’s hard to even make sense of this unless you are thinking very closely about where the primes are. And it takes some thought to see why one derivative is evaluated at u(t), while the other is evaluated just at t.
So how is it that these two expressions are saying the same thing, when they look so different? The key is to understand how the first one (Leibniz form) implicitly involves composition. If \(\scriptsize \frac{{\rm d}y}{{\rm d}u}) is a thing that exists, then y must be a function of u. And if \(\scriptsize \frac{{\rm d}y}{{\rm d}u}) is a thing that exists, then u must be a function of x. In short, x determines the value of u and u determines the value of y, which means that y is also a function of x, whose value is determined by composing the function y and the function u: \(\scriptsize y(x)=y(u(x))\).
Viewed this way, the left side of \(\scriptsize \frac{{\rm d}y}{{\rm d}x}=\frac{{\rm d}y}{{\rm d}u}\cdot\frac{{\rm d}u}{{\rm d}x}\) is the derivative of a composed function. On the right side of this equation, y depends on u and u depends on x, which implies that y depends on x by way of a composition. On the left side, y depends explicitly on x. The two sides together
In practice, I find it quite helpful to have students first use the chain rule in Leibniz notation, explicitly writing composed functions in terms of u. For example, suppose we are given $$y=\sin(x^2).$$ We may split this up as $$y=\sin(u)$$ and $$u=x^2,$$ and students can easily use substitution to confirm that this is correct. Then we have $$\frac{{\rm d}y}{{\rm d}u}=\cos(u),$$ since the \(\scriptsize{\rm d}u\) in the denominator tells us to treat u as a the variable. And similarly, $$\frac{{\rm d}u}{{\rm d}x}=2x,$$ since the \(\scriptsize{\rm d}u\) in the numerator tells us to treat u as a function.
Multiplying these then gives $$\frac{{\rm d}y}{{\rm d}u}=\cos(u)2x,$$ but of course we want our answer (like the function we were given in the first place) to depend on x and not u, so we replace u with \(\scriptsize x^2 \): $$\frac{{\rm d}y}{{\rm d}u}=\cos(x^2)2x.$$
All of this is quite a lot of work compared to the “derivative of the outside times derivative of inside” way of doing the chain rule. And I want students to end up getting comfortable using the chain rule without going through all these steps. But as a tool for learning to use the chain rule it has many advantages:
- It helps students see how the two forms of the chain rule (Leibniz and Prime) are equivalent, and it lets them start taking derivatives using the form that feels more intuitive.
- It makes it clear, from a procedural perspective, why the derivative of \(\scriptsize f(g(x))\) involves composition in one part of the derivative but not the other. Specifically, the derivative of the inner part depends on x, whereas the derivative of the outer part depends on u, which must then be expressed in terms of x.
- It helps students “see” composition and identify inside and outside parts. For example, many students struggle to see $$e^{x^2}$$ as a composed function, and to identify “inside” and “outside” functions. But if you ask: For an appropriate choice of $u$, $$\text{can this be written as }y=e^u \text{ or as } y=u^2 ?$$ most students will quickly see that this function isn’t “something squared” and is “e to something,” meaning that u must be \(\scriptsize x^2\).
\(\scriptsize \) None of this is a magic bullet, but it does get students thinking about how the chain rule operates and how the two forms, seemingly so different, are actually the same.
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