Question: Let $g(x) = e^{f(x)}$ where $f$ is infinitely differentiable function.    Does there exist a formula for $n$-th derivative $g^{(n)}(x)$ where $n\geq 0?$   I obtain the following.  $$g'(x) = f'(x) e^{f(x)}$$ $$g''(x) = f''(x) e^{f(x)} + (f'(x))^2 e^{f(x)}$$ $$g'''(x) = f'''(x) e^{f(x)} + 3 f'(x) f''(x) e^{f(x)} + (f'(x))^3 e^{f(x)}.$$ But I could not obtain any pattern from above.
I want to find the $nth$ derivatives of the function $e^{f(x)}$ with respect to $x$, the first derivative is $$e^{f(x)}f^{\prime}(x).$$  The second derivative is $$\left( {\frac {d^{2}}{d{x}^{2}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+ \left( {\frac {d}{dx}}f \left( x \right)   \right) ^{2}{{\rm e}^{f \left( x \right) }} .$$ The third derivative is $$\left( {\frac {d^{3}}{d{x}^{3}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+3\, \left( {\frac {d^{2}}{d{x}^{2}}}f \left( x  \right)  \right)  \left( {\frac {d}{dx}}f \left( x \right)  \right) { {\rm e}^{f \left( x \right) }}+ \left( {\frac {d}{dx}}f \left( x  \right)  \right) ^{3}{{\rm e}^{f \left( x \right) }} $$  The fourth derivative is $$ \left( {\frac {d^{4}}{d{x}^{4}}}f \left( x \right)  \right) {{\rm e}^ {f \left( x \right) }}+4\, \left( {\frac {d^{3}}{d{x}^{3}}}f \left( x  \right)  \right)  \left( {\frac {d}{dx}}f \left( x \right)  \right) { {\rm e}^{f \left( x \right) }}+3\, \left( {\frac {d^{2}}{d{x}^{2}}}f  \left( x \right)  \right) ^{2}{{\rm e}^{f \left( x \right) }}+6\,  \left( {\frac {d^{2}}{d{x}^{2}}}f \left( x \right)  \right)  \left( { \frac {d}{dx}}f \left( x \right)  \right) ^{2}{{\rm e}^{f \left( x  \right) }}+ \left( {\frac {d}{dx}}f \left( x \right)  \right) ^{4}{ {\rm e}^{f \left( x \right) }} $$  My question is: Is there a general formula or a pattern for the nth derivative of $e^{f(x)}$. You may use the maple command diff(exp(f(x)), x$5) to do some experiments. Thanks a lot ^^
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