Chain Rule Paradox or Am I Doing Something Wrong?

V0ODO0CH1LD

If h(x) = ax, g(x) = bx and f(x) = g(h(x)).

Wouldn't h'(x) = a? And g'(x) = b? And f'(x) = ab?

But the chain rule says f'(x) must equal h'(x)g'(h(x)), so that means f'(x) = ab(ax) = (a^2)bx.

Am I missing something obvious?

jbriggs444

Perhaps.

h'(x) = a. Yes. It's a constant function.
g'(x) = b. Yes. It's a constant function.

The chain rule says that f'(x) = h'(x)g'(h(x)). Yes.

What's g'(h(x)) ?

That's g'(whatever) = b

What's h'(x)?

That's h'(whatever) = a

What's h'(x)g'(h(x)) ?

That's ab, just like it is supposed to be.

mathman

f(x)=g(h(x))=bh(x). f'(x)=bh'(x)=ba. The point is that g'(u)=b, no matter what u is, even if u=h(x).

V0ODO0CH1LD

Okay, but if h(x) = x^2, g(x) = x^2 and f(x) = g(h(x))

Now by your logic f'(x) should be 4x^2 not 4x^3, since h'(x) = 2x and g'(x) = 2x.
Then f'(x) = 2x * 2x = 2x^2.

I am still confused..

jbriggs444

In this case the first derivitives are not constant functions. You need to keep track of what your x's are. Don't let the letters trick you.

Work it through. Chain rule says h'(x)g'(h(x))

h'(x) = 2x.
h(x) = x^2
remember that g'(y) = 2y
so
g'(h(x)) = 2h(x) = 2x^2

Put it together, f'(x) = 2x * 2x^2 = 4x^3

And since f(x) = h(x)^2 = (x^2)^2 = x^4, this result is correct.

V0ODO0CH1LD

So if both equations in the chain are linear I use g'(whatever) and h'(whatever), but if one of them isn't what I take the derivative with respect to matters? Why?

If I use mathman's definition I get that d/dx (x^2)^2 = 4x^2 and if I use jrbriggs444's definition I get that d/dx a(bx) = (b^2)ax. What is wrong?

haruspex

No, it's entirely consistent. You just have to understand that g' here does not mean differentiate g wrt x. It means differentiate g wrt whatever g is a function of, namely h.
I assume you mean h(x) = x², g(u) = u², where u = h.
g'(u) = 2u = 2h(x) = 2x²; h'(x) = 2x; chain rule gives 2x².2x = 4x³
No. jrbriggs444's first post worked this through and got ab. How are you getting (b²)ax?

V0ODO0CH1LD

Thanks, haruspex's post actually cleared my confusion!