From Example Problems
Revision as of 20:49, 21 October 2006 by Geoff (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

f(x)={\frac  {1}{x}}\,

f'(x)=\lim _{{\Delta x\to 0}}{\frac  {f(x+\Delta x)-f(x)}{\Delta x}}=\lim _{{\Delta x\to 0}}{\frac  {{\frac  {1}{x+\Delta x}}-{\frac  {1}{x}}}{\Delta x}}=\lim _{{\Delta x\to 0}}{\frac  {{\frac  {x-(x+\Delta x)}{x(x+\Delta x)}}}{\Delta x}}=\lim _{{\Delta x\to 0}}{\frac  {{\frac  {-\Delta x}{x(x+\Delta x)}}}{\Delta x}}

=\lim _{{\Delta x\to 0}}{\frac  {-\Delta x}{x\Delta x(x+\Delta x)}}=\lim _{{\Delta x\to 0}}{\frac  {-1}{x(x+\Delta x)}}=-{\frac  {1}{x^{2}}}

Thus, we have shown that the derivative of {\frac  {1}{x}} is -1{\frac  {1}{x^{2}}}.

Main Page : Calculus