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Derivative of ln x

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Derivative f’ of the function f(x)=ln x is: f’(x) = 1/x for any positive value of x.


Derivative of natural logarithm ln x

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

$$ \forall x \in ]0, +\infty[ , f’(x) = 1/x$$

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 $\forall x \in ]0, +\infty[$ :

$e^y y’ = e^{f(x)} f’(x) = 1$

because $(u\circ v)’= v’\times u’(v)$ avec $u(x)=e^x$ et $v(x)=f(x)$

By substituting $e^y$ par $x$ we have:

$e^y y’ = x f’(x) = 1$

Then:

$$ f’(x) = 1/x$$

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