Find the derivative of the following function using the limit definition of derivative.

Question

Find the derivative of the following function using the limit definition of derivative. (No credit will be given to any method for finding the derivative except using the limit definition of derivatives)

f (x) = 1 / (1+ √x)

Answer 1

f(x+h) = 1/[ 1 + sqrt(x+h)]

Then

f(x+h) - f(x) =

1/[ 1 + sqrt(x+h)] - 1/(1 + sqrt(x)) =

combines into a single fraction with LCD [(1+sqrt(x+h)) (1+ sqrt(x))]....

[ (1+sqrt(x)) - ( 1 + sqrt(x+h))]/ [ (1 + sqrt(x))(1 + sqrt(x+h))] =

The 1's cancel in the numerator and combines like terms....

[ sqrt(x) - sqrt(x+h)] / [ (1 + sqrt(x))(1 + sqrt(x+h))] =

Rationalizes the numerator by multiplying top and bottom by sqrt(x)+sqrt(x+h).....

[x - (x+h)]/[ (1 + sqrt(x))(1 + sqrt(x+h))(sqrt(x)+sqrt(x+h)] =

The x's cancel in the numerator and combines like terms....

-h / [ (1 + sqrt(x))(1 + sqrt(x+h))(sqrt(x)+sqrt(x+h)]

But...

the difference quotient used in the limit defintion is [ f(x+h)-f(x)] / h

So the H's CANCEL!!!

What remains is:

-1 / [ (1 + sqrt(x))(1 + sqrt(x+h))(sqrt(x)+sqrt(x+h)]

Finally, the limit h-->0 as h tends to zero is:

-1 / {2* [ 1 + sqrt(x)]^2 * sqrt(x)}

This is the derivative

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CHECKS by power rule/chain rule.....

y = (1 + x^(1/2))^(-1)

y' = (-1) * (1 + x^(1/2))^(-2) * (1/2)x^(-1/2)

= -1/ { 2 * [1 + x^(1/2)]^2*sqrt(x)}

which verify and proves the answer is correct