<PRE><B>Question 1204864</B>
**1. Given:**

* f&#39;(g(x)) = g&#39;(x) 
* f&#39;&#39;(x) * f&#39;(x) = f(x)
* g&#39;(3) = 2
* g&#39;&#39;(3) = 2

**2. Find g&#39;&#39;&#39;(3):**

* Differentiate f&#39;(g(x)) = g&#39;(x) with respect to x using the chain rule:
   f&#39;&#39;(g(x)) * g&#39;(x) = g&#39;&#39;(x)

* Substitute x = 3:
   f&#39;&#39;(g(3)) * g&#39;(3) = g&#39;&#39;(3)
   f&#39;&#39;(g(3)) * 2 = 2 
   f&#39;&#39;(g(3)) = 1

**3. Find g&#39;&#39;&#39;&#39;(3):**

* Differentiate f&#39;&#39;(g(x)) * g&#39;(x) = g&#39;&#39;(x) with respect to x using the product rule and chain rule:
   [f&#39;&#39;&#39;(g(x)) * g&#39;(x)] * g&#39;(x) + f&#39;&#39;(g(x)) * g&#39;&#39;(x) = g&#39;&#39;&#39;(x)

* Substitute x = 3:
   [f&#39;&#39;&#39;(g(3)) * 2] * 2 + 1 * 2 = g&#39;&#39;&#39;(3)
   4 * f&#39;&#39;&#39;(g(3)) + 2 = g&#39;&#39;&#39;(3) 

* We need to find f&#39;&#39;&#39;(g(3)). To do this, differentiate the given equation f&#39;&#39;(x) * f&#39;(x) = f(x) with respect to x using the product rule:
   f&#39;&#39;&#39;(x) * f&#39;(x) + f&#39;&#39;(x) * f&#39;&#39;(x) = f&#39;(x)

* Substitute x = g(3) in the above equation:
   f&#39;&#39;&#39;(g(3)) * f&#39;(g(3)) + [f&#39;&#39;(g(3))]^2 = f&#39;(g(3))
   f&#39;&#39;&#39;(g(3)) * 2 + 1^2 = 2 
   2 * f&#39;&#39;&#39;(g(3)) = 1
   f&#39;&#39;&#39;(g(3)) = 1/2

* Now, substitute f&#39;&#39;&#39;(g(3)) = 1/2 back into the equation for g&#39;&#39;&#39;(3):
   4 * (1/2) + 2 = g&#39;&#39;&#39;(3)
   g&#39;&#39;&#39;(3) = 4

**4. Find g&#39;&#39;&#39;&#39;(3):**

* Differentiate the equation for g&#39;&#39;&#39;(x) obtained in step 3:
   [f&#39;&#39;&#39;&#39;(g(x)) * g&#39;(x)] * g&#39;(x) + [f&#39;&#39;&#39;(g(x)) * g&#39;&#39;(x)] * 2 + 2 * g&#39;&#39;&#39;(x) = g&#39;&#39;&#39;&#39;(x)

* Substitute x = 3:
   [f&#39;&#39;&#39;&#39;(g(3)) * 2] * 2 + [1/2 * 2] * 2 + 2 * 4 = g&#39;&#39;&#39;&#39;(3)
   4 * f&#39;&#39;&#39;&#39;(g(3)) + 2 + 8 = g&#39;&#39;&#39;&#39;(3)
   4 * f&#39;&#39;&#39;&#39;(g(3)) + 10 = g&#39;&#39;&#39;&#39;(3)

* To find f&#39;&#39;&#39;&#39;(g(3)), we need to differentiate the equation f&#39;&#39;&#39;(x) * f&#39;(x) + [f&#39;&#39;(x)]^2 = f&#39;(x) with respect to x: 
   f&#39;&#39;&#39;&#39;(x) * f&#39;(x) + f&#39;&#39;&#39;(x) * f&#39;&#39;(x) + 2 * f&#39;&#39;(x) * f&#39;&#39;&#39;(x) = f&#39;&#39;(x)

* Substitute x = g(3) in the above equation:
   f&#39;&#39;&#39;&#39;(g(3)) * 2 + 1/2 * 1 + 2 * 1 * 1/2 = 1 
   2 * f&#39;&#39;&#39;&#39;(g(3)) + 1 = 1
   f&#39;&#39;&#39;&#39;(g(3)) = 0

* Substitute f&#39;&#39;&#39;&#39;(g(3)) = 0 back into the equation for g&#39;&#39;&#39;&#39;(3):
   4 * 0 + 10 = g&#39;&#39;&#39;&#39;(3)
   g&#39;&#39;&#39;&#39;(3) = 10

**5. Find g&#39;&#39;&#39;&#39;&#39;(3):**

* Differentiate the equation for g&#39;&#39;&#39;&#39;(x) obtained in step 4:
   [f&#39;&#39;&#39;&#39;&#39;(g(x)) * g&#39;(x)] * g&#39;(x) + [f&#39;&#39;&#39;&#39;(g(x)) * g&#39;&#39;(x)] * 2 + 2 * g&#39;&#39;&#39;&#39;(x) = g&#39;&#39;&#39;&#39;&#39;(x)

* Substitute x = 3:
   [f&#39;&#39;&#39;&#39;&#39;(g(3)) * 2] * 2 + [0 * 2] * 2 + 2 * 10 = g&#39;&#39;&#39;&#39;&#39;(3)
   4 * f&#39;&#39;&#39;&#39;&#39;(g(3)) + 20 = g&#39;&#39;&#39;&#39;&#39;(3)

* To find f&#39;&#39;&#39;&#39;&#39;(g(3)), we need to differentiate the equation f&#39;&#39;&#39;&#39;(x) * f&#39;(x) + f&#39;&#39;&#39;(x) * f&#39;&#39;(x) + 2 * f&#39;&#39;(x) * f&#39;&#39;&#39;(x) = f&#39;&#39;(x) with respect to x. This will involve higher-order derivatives of f(x) which are not directly provided in the given information. 

**Therefore, we cannot determine the exact value of g&#39;&#39;&#39;&#39;&#39;(3) with the given information.**

**Conclusion:**

The provided information is insufficient to calculate the exact value of (d^5 g)/(dx^5) when x = 3.
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