Derivative of ln x

Derivative of natural logarithm ln x

Derivative $f’$ of the function $f(x)=\ln x$ is:

$$ \forall x \in ]0, +\infty[ , \quad f’(x) = \dfrac{1}{x}$$

Proof

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:

$$ f’(x) = \dfrac{1}{x}$$