Question 1204865: if : f '(g (x)) = g '(x), f ''(x) \[Times] f '(x) = f (x), g '(3) = 2 g ''(3) = 2} then (d ^5 g)/(dx ^5 when x=3 is ... (9 , 10 ,18 , 24)
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! **1. Given:**
* f'(g(x)) = g'(x)
* f''(x) * f'(x) = f(x)
* g'(3) = 2
* g''(3) = 2
**2. Find g'''(3):**
* Differentiate f'(g(x)) = g'(x) with respect to x using the chain rule:
f''(g(x)) * g'(x) = g''(x)
* Substitute x = 3:
f''(g(3)) * g'(3) = g''(3)
f''(g(3)) * 2 = 2
f''(g(3)) = 1
**3. Find g''''(3):**
* Differentiate f''(g(x)) * g'(x) = g''(x) with respect to x using the product rule and chain rule:
[f'''(g(x)) * g'(x)] * g'(x) + f''(g(x)) * g''(x) = g'''(x)
* Substitute x = 3:
[f'''(g(3)) * 2] * 2 + 1 * 2 = g'''(3)
4 * f'''(g(3)) + 2 = g'''(3)
* We need to find f'''(g(3)). To do this, differentiate the given equation f''(x) * f'(x) = f(x) with respect to x using the product rule:
f'''(x) * f'(x) + f''(x) * f''(x) = f'(x)
* Substitute x = g(3) in the above equation:
f'''(g(3)) * f'(g(3)) + [f''(g(3))]^2 = f'(g(3))
f'''(g(3)) * 2 + 1^2 = 2
2 * f'''(g(3)) = 1
f'''(g(3)) = 1/2
* Now, substitute f'''(g(3)) = 1/2 back into the equation for g'''(3):
4 * (1/2) + 2 = g'''(3)
g'''(3) = 4
**4. Find g'''''(3):**
* Differentiate the equation for g'''(x) obtained in step 3:
[f''''(g(x)) * g'(x)] * g'(x) + [f'''(g(x)) * g''(x)] * 2 + 2 * g'''(x) = g''''(x)
* Substitute x = 3:
[f''''(g(3)) * 2] * 2 + [1/2 * 2] * 2 + 2 * 4 = g''''(3)
4 * f''''(g(3)) + 2 + 8 = g''''(3)
4 * f''''(g(3)) + 10 = g''''(3)
* To find f''''(g(3)), we need to differentiate the equation f'''(x) * f'(x) + [f''(x)]^2 = f'(x) with respect to x:
f''''(x) * f'(x) + f'''(x) * f''(x) + 2 * f''(x) * f'''(x) = f''(x)
* Substitute x = g(3) in the above equation:
f''''(g(3)) * 2 + 1/2 * 1 + 2 * 1 * 1/2 = 1
2 * f''''(g(3)) + 1 = 1
f''''(g(3)) = 0
* Substitute f''''(g(3)) = 0 back into the equation for g''''(3):
4 * 0 + 10 = g''''(3)
g''''(3) = 10
**5. Find g'''''(3):**
* Differentiate the equation for g''''(x) obtained in step 4:
[f'''''(g(x)) * g'(x)] * g'(x) + [f''''(g(x)) * g''(x)] * 2 + 2 * g''''(x) = g'''''(x)
* Substitute x = 3:
[f'''''(g(3)) * 2] * 2 + [0 * 2] * 2 + 2 * 10 = g'''''(3)
4 * f'''''(g(3)) + 20 = g'''''(3)
* To find f'''''(g(3)), we need to differentiate the equation f''''(x) * f'(x) + f'''(x) * f''(x) + 2 * f''(x) * f'''(x) = f''(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'''''(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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