I am really struggling with a highschool calculus question which involves finding the derivative of a function using the first principles.

The question is as follows:

Find the derivative of f(x) = (3x-1)/(x+2) when x ≠ -2

I am having trouble with this problem because I am unsure what to do when I have put my function of f(x+h) into the equation f'(x) = [f(x+h)-f(x)]/h.

This is a link so my working so far and I would greatly appreciate it if you could please explain the steps how to finish this question.


Thank you,


  • $\begingroup$ Your working looks fine. Just keep manipulating your limit (you should know how to simplify the addition or subtractions of two fractions) and use the limit laws. $\endgroup$ – mattos Feb 23 '15 at 5:54

With some tedious algebra you have ${f(x+h)-f(x) \over h} = {7 \over x^2+(h+4)x + 2h +4}$, which is 'nicely' behaved at $h=0$, so we see that $\lim_{h \to 0} {f(x+h)-f(x) \over h} = {7 \over x^2+4x + 4} = {7 \over (x+2)^2 }$.

  • $\begingroup$ Thank you very much for this answer. It is very clear and shows the steps. $\endgroup$ – Geoffrey M Feb 23 '15 at 6:10
  • $\begingroup$ Delighted to help! $\endgroup$ – copper.hat Feb 23 '15 at 6:12

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