Induction proof of nth derivative

[Bob Ho]

Show by induction that the nth derivative f⁽ⁿ⁾(x) of;

f(x)=sqrt(1-x)

is

f⁽ⁿ⁾(x)= -\frac{(2n)!}{4^{n}n!(2n-1)}*(1-x)^(1/2)-n

for n \geq 1.

 

[uocan]

For n=1 it works.

For n=k-1 assume that it works. Write n=kth derivative as the derivative of the expression for k-1, i.e d(f(k-1)(x))/(dx). Taking the derivative the result follows.

 

[Redbelly98]

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