Differential of ln(f(x)) proof

 Differential of \(\ln f(x)\)

let \(y = \ln f(x)\)

Taking exponential on both sides:

\[e^{y} = e^{\ln f(x)}\]

but \(e^{\ln f(x)} = f(x)\) since e and ln are inverse function of each other

\[e^{y} = f(x)\]

Differentiating both sides with respect to x

(Remember y is a function of x thus when we differentiate y we ought to multiply with its derivative)

\[e^{y} \dfrac{dy}{dx} = f'(x)\]

\[\dfrac{dy}{dx} = \dfrac{f'(x)}{e^y}\]

but \(e^y = f(x)\)

\[\dfrac{dy}{dx} = \dfrac{f'(x)}{f(x)}\]

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