Question

How would you find the derivative of f(x)=1/sqrtx using limits? I know how to do problems with a fraction or a square root, but I can't figure out how to do one with both. I have tried a few different starts, but always end up getting stuck.

Answer 1

We will be using the binomial expansion
(1+x)ⁿ
=1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...

We will be calculating, for f(x)=1/√x, x≠0, h≠0 and let a=h/x
D(x,h)=(1/√(x+h) - 1/√x)/h
=(1/√x)((1/√(1+a)-1/√1))/h
=(1/√x)(1 + (-1/2)a + (-1/2)(-3/2)a²/2! + ... -1)/h
=-(1/√x)((1/2)(h/x) - (3/8)(h/a)² + ...)/h
=-(1/√x)(1/x - (3/8)(h/x²) + ...)

f'(x) = Lim (h→0)(D(x,h))
=-1/√x)
=-(1/√x)(1/x)
=-x^-3/2

Answer 2

f'(x) = Lim (h→0)(D(x,h))
=-(1/√x)(1/x)
=-x^-3/2