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Derivative f’ of the function natural logarithm f(x)=ln x is: f’(x) = 1/x for any positive value of x.
Derivative $f’$ of the function $f(x)=\ln x$ is:
$$ \forall x \in ]0, +\infty[ , \quad f’(x) = \dfrac{1}{x}$$
Let $y$ the function ln x
$y = f(x)= \ln x$
then by definition (ln is the inverse function of exp)
$e^y = e^{f(x)} = x$
By taking respectively the derivative with respect to $x$ of the two elements, we have $\forall x \in ]0, +\infty[$ :
$e^y y’ = e^{f(x)} f’(x) = 1$
using Chain Rule $(u\circ v)’= v’\times u’(v)$ avec $u(x)=e^x$ and $v(x)=f(x)$
By substituting $e^y$ by $x$ we have:
$e^y y’ = x f’(x) = 1$
Then:
$$ f’(x) = \dfrac{1}{x}$$