Derivative of Tan x
Date: 03/31/2001 at 23:50:29 From: Tong Subject: Derivative of tan x I know that finding derivatives can be found in two ways. One is by taking the limit of (f(x+h)-f(x)) lim h>0 ------------- h Can you show me how to get the derivative of tan x by: tan(x+h)-tan x lim h>0 -------------- h I tried many ways but still can't work it out.
Date: 04/01/2001 at 12:16:29 From: Doctor Jaffee Subject: Re: Derivative of tan x Hi Tong, First of all, the easy way to find the derivative of tan x is to calculate the derivative of sin x/cos x using the quotient rule of derivatives. It works out quickly and easily. However, that is not what you asked, so here is my suggestion for proving that the derivative is the sec^2(x) using the definition of derivative that you are using. As a preliminary step, recall that sin h lim ------ = 1 h->0 h Also, cos 0 = 1 tan x + tan h Furthermore, tan(x + h) = -------------- 1 - tan x tan h So, try again. Use the formula for tan(x + h) that I have above, transform tan h to sin h/cos h, and use the limit of sin h/h that I mentioned above. The result should be that everything cancels out except 1 + tan^2(x), which equals sec^2(x). I hope you find this helpful. If you get stuck, write back and show me everything you have done so far and I'll try to get you back on track. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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