Finding the Derivative by the Limit Process

[forwardthinking22]

Need help on how to find the derivative by the limit process, I've already attempted both but I'm not getting close to what the answer is

1. f(x)=(1)/(x-1)

2. f(x)= squareroot(x+1)

 

[Deleted member 4993]

Need help on how to find the derivative by the limit process, I've already attempted both but I'm not getting close to what the answer is

1. f(x)=(1)/(x-1)

2. f(x)= squareroot(x+1)

Please share your work with us - even if it is incorrect - so that we can assist you in correcting your mistakes.

 

[stapel]

Need help on how to find the derivative by the limit process, I've already attempted both but I'm not getting close to what the answer is

When you reply showing your work, please also specify which form of the "limit process" your book is having you use. Thank you!

 

[DrPhil]

Need help on how to find the derivative by the limit process, I've already attempted both but I'm not getting close to what the answer is

1. f(x)=(1)/(x-1)

2. f(x)= squareroot(x+1)

HINT: both of these involve a form of "rationalization," by which I mean multiplying both the numerator and the denominator by an expression that will help you separate out the "h" term(s), using \(\displaystyle (a - b)(a + b) = a^2 - b^2\).

 

[Deleted member 4993]

The first one does not "require" rationalization:

\(\displaystyle \displaystyle f(x) \ = \ \frac{1}{x - 1}\)

\(\displaystyle \displaystyle f(x + h) \ = \ \frac{1}{x + h - 1}\)

\(\displaystyle \displaystyle f(x + h) - f(x)\ = \ \frac{-h}{(x + h - 1)(x - 1)}\)

\(\displaystyle \displaystyle \frac{f(x + h) - f(x)}{h}\ = \ \frac{-1}{(x + h - 1)(x - 1)}\)

\(\displaystyle \displaystyle \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}\ = \ \frac{-1}{(x - 1)(x - 1)}\)

\(\displaystyle \displaystyle \frac{d}{dx} f(x)\ = \ -\frac{1}{(x - 1)^2}\)