Skip to content

Lars Eighner's Homepage


LarsWiki

calumus meretrix et gladio innocentis

Derivative of a Function at a Point

This a demonstration of two ways of finding the derivative of a function {$f(x)$} at a point {$(x_0,f(x_0))$}.

For this demonstration, the function that will be used is

{$$ f(x) = \sqrt{x+1} $$}

and the point used will be {$(3,2)$}, which can be verified to be a point on the function.

Solution directly from the definition

We use the basic definition of the derivative:

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

But at the particular point, which we know to be on the function, we know {$ x=3 $} and {$ f(x) = 2). As in most case, it is easier to express a root as an exponent, so

{$$ f(3+h) = \sqrt{(3+h)+1} = ((3+h) + 1)^{1 \over 2} = (4+h)^{1\over 2} $$}

{$$ \begin{align} f^\prime (3) &= \lim_{ h \rightarrow0} {{f(3 + h) -f(3)} \over { h}} \cr &= \lim_{ h \rightarrow0} {{(4+h)^{1\over 2} -2} \over { h}} \cr &= \lim_{ h \rightarrow0} {{[(4+h)^{1\over 2} -2][(4+h)^{1\over 2} +2]} \over { h[(4+h)^{1\over 2} +2]}} \cr &= \lim_{ h \rightarrow0} {{[(4+h)^{1\over 2}]^2 -4} \over { h[(4+h)^{1\over 2} +2]}} \cr &= \lim_{ h \rightarrow0} {{(4+h) -4} \over { h[(4+h)^{1\over 2} +2]}} \cr &= \lim_{ h \rightarrow0} {h \over { h[(4+h)^{1\over 2} +2]}} \cr &= \lim_{ h \rightarrow0} {1 \over { (4+h)^{1\over 2} +2}} \cr &= {1 \over { (4+0)^{1\over 2} +2}} \cr f^\prime (3) &= {1 \over 4} \end{align} $$}


Plug into the derivative function

The first method may be a large amount of work to invest on just one point of just one function. You may know a general formula for finding the derivative function quickly.

Powers Rule

{$$ {d \over dx} x^n = nx^{n-1} $$}

With the power rule it is very easy to find the function for the derivative.

{$$ f(x) = \sqrt{x+1} = (x+1)^{1 \over 2} $$}

so letting {$ n = {1 \over 2}$} and applying the powers rule:

{$$ f'(x) = {1 \over 2} (x+1)^{{1 \over 2} -1} = {1 \over 2} (x+1)^{-{1\over 2}} $$}

Then by substituting the value of x at (3,2) in the derivative function, the derivative at the point is found:

{$$ f'(3) = {1 \over 2} (3+1)^{-{1\over 2}} = ({1 \over 2})({1 \over 2}) = {1 \over 4} $$}

However, the derivative function can also be obtained directly from the definition of derivative.

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

Then the x value of the point can be plugged in to find the derivative at the point.

{$$ f^\prime (3) = {1 \over {2 \sqrt{3+1} } }= {1 \over 4} $$}


Graph: FooPlot Online graphing calculator and function plotter

Sources:

Recommended:

Category: Math


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 MathOlivia

August 05, 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