Hey everybody please help, i have no clue how to start this one. Could you please show any intermediate steps.
Show by induction that the nth derivative f^n(x) of
f(x) = sqrt(1-x)
f^n(x) = -(2n)!/((4^n)n!(2n-1)) * (1-x)^((1/2)-n)
for n>= 1
Would you differentiate the equation with respect to x?? Would this just not make the first part of the equation a constant??
SO pretty much how do you differentiate it?
P.s. All other induction that i have done has involved a sequence, is this not necessary for all cases.
good ol' chain ruleSO pretty much how do you differentiate it?
doesn't matter. the method of prove by induction is the same, no matter what you are working withP.s. All other induction that i have done has involved a sequence, is this not necessary for all cases.
Awesome that was really helpful thanks.
I've tried to simplify as far as possible but i got.
2(2n)!/4^(n+1)n! * (1-x)^((1/2)-(n+1)) Equation 1
Does this seem kind of right??
It's obviously not what i expected, as it should equal the original equation just with (n+1) in place of the n's.
Intermediate steps for the red part of the equation i just got where:
2(4n^3 + 4n^2 - n - 1)(2!)/ 4(4n^3 + 4n^2 - n -1)4^n(n!)
canceling the orange parts, gave me the equation 1.
a bit more of a nudge in the right direction: change the top this way. split the first 4 into 2*2, use one two to multiply the (n + 1) to get (2n + 2) and the other 2 to multiply the (n - 1/2) to get (2n - 1)
so you have: