What is the derivative of ln(1/x+ln(1/x+ln(1/x))) ?

The derivative of ln(1/x+ln(1/x+ln(1/x))) is (xln(x)-1-x-x^2)/(x(-xln(x)+1)(1+xln(1/x-ln(x))))

What is the first derivative of ln(1/x+ln(1/x+ln(1/x))) ?

The first derivative of ln(1/x+ln(1/x+ln(1/x))) is (xln(x)-1-x-x^2)/(x(-xln(x)+1)(1+xln(1/x-ln(x))))

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

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

Solution steps

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

Simplify

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

n = 1 \sum \infty n e^{- n^{2}}

\int \frac{6 x}{( x ^{2} + 5 ) ^{2}} d x

y^{^{''}} + y^{^{'}} - 6 y = 0, y (0) = 10, y^{^{'}} (0) = 0

\int_{0}^{\frac{π}{4}} sin (2 x) d x

\frac{\partial}{\partial x} (\frac{( 2 x - y )}{x y})

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