What a jerk!
A jerk is defined to be the third derivative of a position. Let denote the function of the infinite power tower as described above. Evaluate the jerk of this function at .
The answer is 9.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
y = x y ⇒ l n y = y l n x d i f f e r e n t i a t i n g w . r . t x ⇒ y y 1 = y 1 l n x + x y ⇒ y 1 = y 1 ( y l n x ) + x y 2 = y 1 l n y + x y 2 . . . ( i ) d i f f e r e n t i a t e a g a i n , ⇒ y 2 = y 2 l n y + y ( y 1 ) 2 + x 2 2 x y y 1 − y 2 . . . ( i i ) d i f f e r e n t i a t e a g a i n , ⇒ y 3 = ( y 3 l n y + y 2 y y 1 ) + y 2 2 y y 1 y 2 − ( y 1 ) 3 + ( x 2 ) 2 x 2 { 2 y y 1 + 2 x ( y 1 ) 2 + 2 x y y 2 − 2 y y 1 } − 2 x ( 2 x y y 1 − y 2 ) . . . ( i i i ) i f x = 1 ⇒ y ( 1 ) = 1 B y e q n ( i ) , y 1 ( 1 ) = y 1 ( 1 ) l n 1 + 1 1 2 = 1 B y e q n ( i i ) , y 2 ( 1 ) = y 2 ( 1 ) l n 1 + 1 ( 1 ) 2 + 1 2 2 × 1 × 1 × 1 − 1 2 = 2 B y e q n ( i i ) , y 3 ( 1 ) = ( y 3 ( 1 ) l n 1 + 2 × 1 1 ) + 1 2 2 × 1 × 1 × 2 − ( 1 ) 3 + ( 1 2 ) 2 1 2 { 2 × 1 × 1 + 2 × 1 × 1 2 + 2 × 1 × 1 × 2 − 2 × 1 × 1 } − 2 × 1 ( 2 × 1 × 1 × 1 − 1 2 ) = 2 + 1 4 − 1 + 1 ( 2 + 2 + 4 − 2 ) − 2 ( 2 − 1 ) = 9