Skip to content

Lars Eighner's Homepage


LarsWiki

calumus meretrix et gladio innocentis

Derivative of 1/x from Definition.


f(x) = 1/x
Contents

This derivative is one of the worked examples in the MIT Open Course Ware cited below. All I am really doing here is reworking in a different notation. In Geometric Interpretation of Derivative I looked at this function and found the derivative at the particular point P = (2/3, 3/2). Here we find the derivative for the general x wherever f(x) is defined.

This is really a special case of the Powers Rule:

{$$ f'(x^n) = {d \over dx} x^n = nx^{n-1} $$}

because {$ 1/x = x^{-1} $}.

Definition of the Derivative

The definition of derivative I will use is:

{$$ f^\prime (x) = \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over { h}} $$}

which differs only in notation from other definitions.

Solution

Plug into the definition:

{$$ \begin{align} f^\prime (x) &= \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over { h}} \cr &= \lim_{ h \rightarrow0} {{ 1/(x + h) - 1/x} \over { h}} \end{align} $$}

As always in derivatives the limit cannot be solved by substitution because the denominator would become 0.

The next step is to express the fractions in the denominator with a common denominator:

{$$ \begin{align} f^\prime (x)&= \lim_{ h \rightarrow0} {{ 1/(x + h) - 1/x} \over { h}} \cr &= \lim_{ h \rightarrow0} {{ (x - (x +h))/(x(x + h)) } \over { h}} \cr &= \lim_{ h \rightarrow0} {{ -h /(x^2 + xh) } \over { h}} \cr &= \lim_{ h \rightarrow0} {{ h (-1 /(x^2 + xh)) } \over { h}} \end{align} $$}

Now h in the denominator cancels h in the numerator, and the limit can be solved by substitution.

{$$ \begin{align} f^\prime (x)&= \lim_{ h \rightarrow0} {{ h (-1 /(x^2 + xh)) } \over { h}} \cr &= \lim_{ h \rightarrow0} {{ -1 } \over {x^2 + xh }} \cr &= \frac{ -1 }{x^2 + x(0) } \cr \therefore f^\prime (x) &= -1 \left( \frac{ 1 }{x^2 } \right) = -1x^{-2} \end{align} $$}

This of course is undefined when x=0, but so is the original function.


Sources:

  1. MIT OpenCourseWare http://ocw.mit.edu "18.01SC Single Variable Calculus: Fall 2010" Creative Commons License Terms. Session 2: Examples of Derivatives | Part A: Definition and Basic Rules | 1. Differentiation | Single Variable Calculus | Mathematics | MIT OpenCourseWare
  2. Example 1. f(x) = 1/x - MIT18_01SCF10_Ses2a.pdf

Recommended:

Category: Math Calculua


Read or Post Comments

No comments yet.

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

Figures are often enhanced by hand editing; the same results may not be achieved with source sites and source apps.

Backlinks

This page is MathCalculusSingleVariableDerivativeOf1OverXFromDefinition

August 06, 2017

  • HomePage
  • WikiSandbox

Lars

Contact by Snail!

Lars Eighner
APT 1191
8800 N IH 35
AUSTIN TX 78753
USA

Help

HOME

The best way to look for anything in LarsWiki is to use the search bar.

Page List

Categories

Physics Pages

Math Pages

Math Exercises

Math Tools

Sections