isi-entrance 2015 Q11

isi-entrance · India · UGA 4 marks Chain Rule Iterated/Nested Exponential Differentiation
Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:
(a) $f _ { n } ( x )$
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.
(c) Use induction and chain rule of differentiation.
Let $f _ { 1 } ( x ) = e ^ { x } , f _ { 2 } ( x ) = e ^ { f _ { 1 } ( x ) }$ and generally $f _ { n + 1 } ( x ) = e ^ { f _ { n } ( x ) }$ for all $n \geq 1$. For any fixed $n$, the value of $\frac { d } { d x } f _ { n } ( x )$ is:\\
(a) $f _ { n } ( x )$\\
(b) $f _ { n } ( x ) f _ { n - 1 } ( x )$\\
(c) $f _ { n } ( x ) f _ { n - 1 } ( x ) \ldots f _ { 1 } ( x )$\\
(d) $f _ { n + 1 } ( x ) f _ { n } ( x ) \ldots f _ { 1 } ( x ) e ^ { x }$.