Question

Oh shoot, I mistyped my previous problem :-/.

f(x)=x^-1/2. Form and simplify into {f(x+h) - f(x)}/h

Answer 1

\[f(x)= 1/\sqrt{x}\] just in case there's any confusion

Answer 2

this is definition of derivative using limits
\[\large \lim_{h \rightarrow 0}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}\]
combine fractions on top, simplify into single fraction
\[\large \lim_{h \rightarrow 0}\frac{\sqrt{x}-\sqrt{x+h}}{h \sqrt{x+h}\sqrt{x}}\]
multiply top/bottom by conjugate
\[\large \lim_{h \rightarrow 0}\frac{\sqrt{x}-\sqrt{x+h}}{h \sqrt{x+h}\sqrt{x}}*\frac{\sqrt{x}+\sqrt{x+h}}{\sqrt{x}+\sqrt{x+h}}\]
\[\large \lim_{h \rightarrow 0}\frac{-h}{h \sqrt{x+h}\sqrt{x}(\sqrt{x}+\sqrt{x+h})}\]
cancel the h's
evaluate at h=0
\[= -\frac{1}{2x \sqrt{x}}\]