What is the derivative of $ln (\frac{1}{x})$ $ln(1/x)$ ?

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Answer 1 · 2015-12-25T13:49:02

We'll need the chain rule here, which states that $\frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x}$ $(dy)/(dx)=(dy)/(du)(du)/(dx)$

In this case, we must rename $u = (\frac{1}{x})$ $u=(1/x)$ and now derivate the function $y = ln u$ $y=lnu$ . Let's do it separately, step-by-step:

$\frac{d y}{d u} = \frac{1}{u}$ $(dy)/(du)=1/u$

$\frac{d u}{d x} = - \frac{1}{x^{2}}$ $(du)/(dx)=-1/x^2$

$\frac{d y}{d x} = - \frac{1}{u x^{2}}$ $(dy)/(dx)=-1/(ux^2)$

Substituting $u$ $u$ :

$\frac{d y}{d x} = - \frac{1}{(\frac{1}{x}) x^{2}} = - \frac{1}{x}$ $(dy)/(dx)=-1/((1/x)x^2)=-1/x$

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