Expert Answer to "Find the derivatives of the functions. f(x)=1xln⁡x"

Clifland

Answered question

2021-05-17

Find the derivatives of the functions.

f (x) = \frac{1}{x \ln x}

Clara Reese

Skilled2021-05-18Added 120 answers

The function we want to differentiate is

f (x) = \frac{1}{x \ln x}

Apply the quotient rule first to get

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

Note that the first term in the numerator is zero because

\frac{d}{d x} (1) = 0

The second term can be computed using the product rule and the derivative of natural logarithm rule:

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

So we get

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

= \ln x + x \frac{1}{x}

= \ln x + 1

Substituting back, we get

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

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