We are given a composite function $r(x) = f(g(h(x)))$, and we are tasked with finding $r'(1)$. The values provided are:

• $h(1) = 3$
• $g(3) = 2$
• $h'(1) = 9$
• $g'(3) = -2$
• $f'(2) = 5$

Step-by-Step Solution

We can use the chain rule for derivatives. For the composite function $r(x) = f(g(h(x)))$, the derivative $r'(x)$ is calculated as:

$r'(x) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$

Now, let's evaluate this at $x = 1$:

1. Find $h(1)$: We are given $h(1) = 3$, so $h(1) = 3$.

2. Find $g(h(1)) = g(3)$: We are given $g(3) = 2$, so $g(h(1)) = g(3) = 2$.

3. Find $f'(g(h(1))) = f'(2)$: We are given $f'(2) = 5$, so $f'(g(h(1))) = f'(2) = 5$.

4. Find $g'(h(1)) = g'(3)$: We are given $g'(3) = -2$, so $g'(h(1)) = g'(3) = -2$.

5. Find $h'(1)$: We are given $h'(1) = 9$, so $h'(1) = 9$.

Now, put everything together:

$r'(1) = f'(g(h(1))) \cdot g'(h(1)) \cdot h'(1)$

Substitute the known values:

$r'(1) = 5 \cdot (-2) \cdot 9$

Simplifying:

$r'(1) = 5 \cdot (-18) = -90$

Thus, $r'(1) = -90$.

Let me know if you would like further details or have any questions!

Here are 5 related questions to expand your understanding:

1. What is the chain rule and how is it applied to composite functions?
2. How would you differentiate a composite function if more than three nested functions were involved?
3. If $f(x)$, $g(x)$, and $h(x)$ were trigonometric functions, how would the process change?
4. How do you interpret the physical meaning of a derivative in such composite functions?
5. How would the result change if one of the derivative values was zero?

Tip: Always write down each step of substitution clearly when working with chain rule problems—it helps avoid mistakes with nested derivatives.