Quotient Rule for Derivatives/Examples/Exponential of x over x

Example of Use of Quotient Rule for Derivatives

$\map {\dfrac \d {\d x} } {\dfrac {e^x} x} = \dfrac {e^x \paren {x - 1} } {x^2}$

Proof

\(\ds \map {\dfrac \d {\d x} } {\dfrac {e^x} x}\)

\(=\)

\(\ds \dfrac {x \map {\frac \d {\d x} } {e^x} - e^x \map {\frac \d {\d x} } x} {x^2}\)

Quotient Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac {x \cdot e^x - e^x \cdot 1} {x^2}\)

Derivative of Exponential Function, Derivative of Identity Function

\(\ds \)

\(=\)

\(\ds \dfrac {e^x \paren {x - 1} } {x^2}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $26$.