MarkT Posted August 28, 2016 Share Posted August 28, 2016 My DS is studying Calc 1. He is doing finding derivatives by the limits method: f'(x) = lim f(x+deltax) - f(x) deltax->0 ------------------- deltax problem: 7 f(x) = ------- srqt(2x) after pulling out the constant to save for later 1 1 ------ - --- srqt((x+deltax) srqt(x) ---------------------------------- deltax after rationalizing this mess I still have 0/0 indeterminate form ?? (tried several approaches) this is really a rational radical expressions problem (rusty) can someone take a crack at this - show steps (posting a picture or via PM since math style above is very painful) the final answer is easy to figure out the normal derivative way not looking for that thank you Quote Link to comment Share on other sites More sharing options...
regentrude Posted August 28, 2016 Share Posted August 28, 2016 Can you do it by using the inverse function rule? Instead of finding the limit of the quotient for 1/sqrt(x) you'd have to do it for 1/y^2. Quote Link to comment Share on other sites More sharing options...
regentrude Posted August 28, 2016 Share Posted August 28, 2016 It works directly, too. set up the quotient, put on common denominator, square the equation that leaves an expression that contains one sqrt move all other terms to the left side to the derivative and square again to get rid of the sqrts. this gives you a quadratic equation for the derivative which resolves nicely 1 Quote Link to comment Share on other sites More sharing options...
Cosmos Posted August 29, 2016 Share Posted August 29, 2016 What went wrong when you rationalized? The trick here is to multiply the numerator and denominator by the conjugate of the numerator. (Often in rational expressions, you multiply by the conjugate of the denominator but not here.) You start with (I'm using h instead of deltax for ease of writing): 1/h ( 1/sqrt(x+h) - 1/sqrt(x) ) Put over one denominator: ( sqrt(x) - sqrt(x+h) ) / ( h sqrt(x) sqrt(x+h) ) Now multiply top and bottom by sqrt(x) + sqrt(x+h): ( x - (x+h) ) / ( h sqrt(x) sqrt(x+h) (sqrt(x) + sqrt(x+h)) ) Top simplifies to h, which cancels with the h in the bottom to give: 1 / (sqrt(x) sqrt(x+h) (sqrt(x) + sqrt(x+h)) And you can easily take the limit of that expression. 1 Quote Link to comment Share on other sites More sharing options...
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