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Popular Calculus >

derivative of 1/(1+1/x)

Solution

\frac{d}{d x} (\frac{1}{1 + \frac{1}{x}})

Solution

\frac{1}{( x + 1 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{1}{1 + \frac{1}{x}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} ((1 + \frac{1}{x})^{- 1})

Apply the chain rule:

- \frac{1}{( 1 + \frac{1}{x} ) ^{2}} \frac{d}{d x} (1 + \frac{1}{x})

= - \frac{1}{( 1 + \frac{1}{x} ) ^{2}} \frac{d}{d x} (1 + \frac{1}{x})

\frac{d}{d x} (1 + \frac{1}{x}) = - \frac{1}{x ^{2}}

= - \frac{1}{( 1 + \frac{1}{x} ) ^{2}} (- \frac{1}{x ^{2}})

Simplify

- \frac{1}{( 1 + \frac{1}{x} ) ^{2}} (- \frac{1}{x ^{2}}) : \frac{1}{( x + 1 ) ^{2}}

= \frac{1}{( x + 1 ) ^{2}}

Graph

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Frequently Asked Questions (FAQ)

What is the derivative of 1/(1+1/x) ?
The derivative of 1/(1+1/x) is 1/((x+1)^2)
What is the first derivative of 1/(1+1/x) ?
The first derivative of 1/(1+1/x) is 1/((x+1)^2)