If you are using first principles to find a derivative, functions with radicals are usually simplified by rationalising the numerator.
I'm not good at fractions and radicals and I have two questions that I can't answer. Here is the questions
a. sqrt[ (3x+1) / (2x-1)]
b. 2 / [ (6x^2+5x+1) ]^2
How can I solve this using this formula f(x) = [ f(x+h) - f(x) ] / h . I will really appreciate if you can give me the full solution + explanation (process). Thanks.
Here is (a):
From line 1 to 2, I rationalized the denominator.
From line 2 to 3, I found a common denominator and instead of dividing the whole thing by h, I multiplied by 1/h.
On the third line, I multiplied top and bottom by the conjugate of the top.
From line 3 to 4, I used the fact that .
From line 4 to 5, I simplified the numerator.
From line 5 to 6, I cancelled from both the numerator and denominator.
From line 6 to 7, I plugged in h=0 and collected like terms.