# Math Help - Derivative of 1/X and root X

1. ## 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

2. 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 $\lim_{\delta \rightarrow 0} \frac{f(x+\delta) - f(x)}{\delta}$.

3. 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

4. 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.

5. 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

6. 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 $f(x) = \frac{1}{x}$

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

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

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

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

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

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

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