Derivative of 1/X and root X

• Sep 8th 2008, 03:28 PM
stones44
Derivative of 1/X and root X
can someone do these step by step please..im new at calculus and dont have my textbook with me

so far i have (A = delta)

root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

and 1/x+Ax - 1/x / (x+Ax) -x

i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
thanks
• Sep 8th 2008, 03:44 PM
mr fantastic
Quote:

Originally Posted by stones44
can someone do these step by step please..im new at calculus and dont have my textbook with me

so far i have (A = delta)

root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

and 1/x+Ax - 1/x / (x+Ax) -x

i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
thanks

Let $f(x) = \sqrt{x}$.

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

$f(x + \delta) - f(x) = \sqrt{x + \delta} - \sqrt{x}$

$\frac{f(x+\delta) - f(x)}{\delta} = \frac{\sqrt{x + \delta} - \sqrt{x}}{\delta} = \frac{(\sqrt{x + \delta} - \sqrt{x})(\sqrt{x + \delta} + \sqrt{x})}{\delta (\sqrt{x + \delta} + \sqrt{x})}$

$= \frac{\delta}{\delta (\sqrt{x + \delta} + \sqrt{x})} = \frac{1}{\sqrt{x + \delta} + \sqrt{x}}$.

I'm sure you can do the final step and find $\lim_{\delta \rightarrow 0} \frac{f(x+\delta) - f(x)}{\delta}$.
• Sep 8th 2008, 04:13 PM
stones44
im sorry im still kind of confused...you just put in part of that equation? the $(\sqrt{x + \delta} + \sqrt{x})$ part and if so, why? its seems kind of random to add that whole part

and what about the 1/x
• Sep 8th 2008, 04:26 PM
Chop Suey
Quote:

Originally Posted by stones44
im sorry im still kind of confused...you just put in part of that equation? the $(\sqrt{x + \delta} + \sqrt{x})$ part and if so, why? its seems kind of random to add that whole part

He multiplied 1 using the conjugate of $\sqrt{x + \delta} - \sqrt{x}$. This enables us to cancel the $\delta$ in the denominator and then we can evaluate the limit.

Follow the same procedure as Mr Fantastic's post for $\frac{1}{x}$. You will need to combine fractions.
• Sep 8th 2008, 04:43 PM
stones44
ok i got it down to
( 1 / (x^2 + (Dx)^2) ) - ( 1 / x^2 ) / Dx( (1/x+Dx) + 1/x )

how to i combine the numerator
• Sep 8th 2008, 05:04 PM
Shyam
Quote:

Originally Posted by stones44
can someone do these step by step please..im new at calculus and dont have my textbook with me

so far i have (A = delta)

root(x+Ax) - rootx / (x+Ax) - x and ive tried many different methods but i mess up

and 1/x+Ax - 1/x / (x+Ax) -x

i tried doing it but i keep getting stuck..and please dont use laws or anything..just algebra cause i just started
thanks

Let $f(x) = \frac{1}{x}$

$f(x+ \Delta x) = \frac {1}{x+ \Delta x}$

$f(x+ \Delta x) -f(x) = \frac {1}{x+ \Delta x}- \frac{1}{x}$

$= \frac {x-(x+ \Delta x)}{x(x+ \Delta x)}$

$= \frac {- \Delta x}{x(x+ \Delta x)}$

$\lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}=\lim_{\Delta x \to 0} \frac {- \Delta x}{x \Delta x(x+ \Delta x)}$

$=\lim_{\Delta x \to 0} \frac {- 1}{x (x+ \Delta x)}$

$= \frac{-1}{x^2}$