How do I Proof with induction that(d/dx)x^(n/2) = (n/2)x^((n/2)-1) ? for 

Expert Answers
lfryerda eNotes educator| Certified Educator

To prove the power rule `d/{dx}x^{n/2}=n/2 x^{n/2-1}` using induction, we first need to prove that it is true for n=1.

Since `d/{dx}x^{1/2}=1/2x^{-1/2}` , then LS=RS so this is true for n=1.

Now assume that the statement is true for n=k.  That is, assume that `d/{dx}x^{k/2}=k/2x^{k/2-1}` is true.

Now we need to use our assumed statement to show that LS=RS of the statement `d/{dx}x^{{k+1}/2}={k+1}/2x^{{k+1}/2-1}={k+1}/2x^{{k-1}/2}` .

`LS=d/{dx}x^{{k+1}/2}`     split up the exponents

`=d/{dx}(x^{k/2}x^{1/2})`    now use product rule

`=k/2x^{k/2-1}x^{1/2}+1/2x^{k/2}x^{-1/2}`   using the assumed statement.  now simplify

`=k/2x^{k/2}x^{-2/2}x^{1/2}+1/2x^{k/2}x^{-1/2}`   factor

`=1/2x^{k/2}(kx^{-1/2}+x^{-1/2})`   factor again




The statement has been proven using induction.