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anonymous
 one year ago
What are the steps to get the derivative of y= (1/x+1)+x+1
anonymous
 one year ago
What are the steps to get the derivative of y= (1/x+1)+x+1

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[y=\left(\frac{1}{x+1}\right) +x+1\] First we can take the stuff in the parenthesis and bring the stuff in the denominator to the numerator so we can use the power rule + chain rule instead of the quotient rule. \[y=(x+1)^{1}+x+1\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now we just take the derivative of each term with respect to x. For the first term we'll use the power rule: \(\frac{d}{dx} (x^n) = n \cdot x^{n1}\) where n is the pwoer of the term

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So the derivative of the first term, \((x+1)^{1}\), using the power rule + chain rule, would be : \[(x+1)^{11} = (x+1)^{2}\] Now the next term, \(x\) . Taking the derivative of \(x\) with respect to \(x\) is just \(1\). \[\frac{d}{dx} (x^1) = 1 \cdot x^{11} = 1 \cdot x^0 =1\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Lastly, the derivative of a constant is always \(0\), therefore \(\dfrac{d}{dx} (1) = 0\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Putting it all together, we have: \[\frac{dy}{dx} = (x+1)^{2} +1 +0\]\[\frac{dy}{dx} = \frac{1}{(x+1)^2} +1\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I forgot to mention the chain rule part of \(\frac{d}{dx}(x+1)^{−1}\). Upon using the power rule, you apply the chain rule to the inner most function, which in this case would be x. Therefore: \[\frac{d}{dx}(x+1)^{1}\color{red}{=} (x+1)^{2} \cdot \frac{d}{dx} (x) = (x+1)^{2} \cdot (1) = (x+1)^{2}\]
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