What is (\partial)/(\partial x)(ln(1+1/x)) ?

The answer to (\partial)/(\partial x)(ln(1+1/x)) is -1/(x(x+1))

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(\partial)/(\partial x)(ln(1+1/x))

\frac{\partial}{\partial x} (ln (1 + \frac{1}{x}))

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

Solution steps

\frac{\partial}{\partial x} (ln (1 + \frac{1}{x}))

Apply the chain rule:

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

= \frac{1}{1 + \frac{1}{x}} \frac{\partial}{\partial x} (1 + \frac{1}{x})

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

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

Simplify

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

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

\frac{\partial}{\partial x} (x^{2} + 2 x y + 3 y^{2})

\int \frac{1}{u} d u

y^{^{''}} - 2 y^{^{'}} - 15 y = 0

(4 y + y x^{2}) d y = (2 x + x y^{2}) d x

\frac{d}{d x} (8 e^{2 x} x + 12 e^{2 x})

What is (\partial)/(\partial x)(ln(1+1/x)) ?
The answer to (\partial)/(\partial x)(ln(1+1/x)) is -1/(x(x+1))