Question 1201323
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Whenever dealing with complicated formulas, it helps to break things up into smaller pieces.


*[tex \Large w(\text{x}) = \frac{f(\text{x})}{g(\text{x})}]
where
*[tex \Large f(\text{x}) = e^{\text{x}}-11\ln(\text{x})+340\text{x}^5]
*[tex \Large g(\text{x}) = 7\text{x}^3-32e^{\text{x}}+5\ln(\text{x})]


We'll need the derivatives of the functions f(x) and g(x) separately.
*[tex \Large f'(\text{x}) = e^{\text{x}}-\frac{11}{\text{x}}+1700\text{x}^4]
*[tex \Large g'(\text{x}) = 21\text{x}^2-32e^{\text{x}}+\frac{5}{\text{x}}]
Let me know if you need me to go over the derivative rules.


I'll rewrite each fraction of the form a/b into ab^(-1)
*[tex \Large f'(\text{x}) = e^{\text{x}}-11\text{x}^{-1}+1700\text{x}^4]
*[tex \Large g'(\text{x}) = 21\text{x}^2-32e^{\text{x}}+5\text{x}^{-1}]



Use the quotient rule to apply the derivative to w(x)
*[tex \Large w(\text{x}) = \frac{f(\text{x})}{g(\text{x})}]


*[tex \Large w'(\text{x}) = \frac{f'(\text{x})g(\text{x}) - f(\text{x})g'(\text{x})}{(g(\text{x}))^2}]


*[tex \Large w'(\text{x}) = \frac{(e^{\text{x}}-11\text{x}^{-1}+1700\text{x}^4)(7\text{x}^3-32e^{\text{x}}+5\ln(\text{x}))-(e^{\text{x}}-11\ln(\text{x})+340\text{x}^5)(21\text{x}^2-32e^{\text{x}}+5\text{x}^{-1})}{(7\text{x}^3-32e^{\text{x}}+5\ln(\text{x}))^2}]


Things spiral out of hand by the third step.
You can optionally expand things out and combine like terms. 
But it's probably better to leave it as it is.
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