Differential of e^{ax} proof

 Differential of \( e^{ax}\)

let \( y = e^{ax}\)

Taking the natural log on both sides:

\[\ln y = \ln e^{ax}\]

Apply one of the logarithm rules:

\[\ln y = ax \ln e \]

but \[\ln e = 1 \]

\[\ln y = ax\]

Taking the differential with respect to x:

(Remember: y is a function of x therefore after differentiating we multiply by the derivative)

\[\dfrac{1}{y} \dfrac{dy}{dx} = a \]

\[\dfrac{dy}{dx} = ay\]

but \(y = e^{ax}\)

\[\dfrac{dy}{dx} = ae^{ax}\]  

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