Question

Let r(x)=f(g(h(x))), where h(1)=2, g(2)=2, h′(1)=8, g′(2)=−2, and f′(2)=5. Find r′(1).

Answer 1

This will require multiple applications of the chain rule. I'm assuming you've recently covered the chain rule?

Answer 2

yup

Answer 3

Excellent. Since we're looking for r'(1), we need to find an expression for r'(x). So for now, consider g(h(x)) as a single function. Then\[\frac{d}{dx}f(g(h(x)))=f'(g(h(x)))\cdot\frac{d}{dx} g(h(x))\]Now we apply the chain rule again to get\[f'(g(h(x)))g'(h(x))h'(x).\]Following this so far?

Answer 4

no not really

Answer 5

Alright. Instead of writing g(h(x)), let's just write k(x), where k(x)=g(h(x)). Then \(f(g(h(x)))=f(k(x))\). This expression we may apply the chain rule to. This gets us to\[\frac{d}{dx}f(k(x))=f'(k(x))\cdot k'(x).\]Now, we need to expand \(k(x)=g(h(x))\). So using the chain rule again, \[\frac{d}{dx}k(x)=\frac{d}{dx}g(h(x))=g'(h(x))\cdot h'(x).\]Did this help clear things up, or are you still stuck on something?

Answer 6

ok so we have this. f′(g(h(x)))g′(h(x))h′(x). what do i do with this

Answer 7

Now, you plug in 1 for x, and slowly simplify everything.

Answer 8

So \[r'(1)=f′(g(h(1)))\cdot g′(h(1))\cdot h′(1)\]The first thing we need to evaluate, is h(1) and h'(1). But we were given h(1)=2, and h'(1)=8. So\[r'(1)=f'(g(2))\cdot g'(2)\cdot8.\]That step make sense?

Answer 9

yeah

Answer 10

Now, we have g(2), and g'(2). But again, these were given to us! So we can simplify even further to get\[r'(1)=f'(2)\cdot-2\cdot 8,\]And do the same thing one more time with f'(2) to get\[r'(1)=5\cdot-2\cdot8=-80\]

Answer 11

Hopefully you followed all that. Let me know if you have any more questions about this.