1. ## Proving the derivative.

"Use the limit definition of the derviative to prove the derivative for the function y = tan x is sec^2 x

(You'll need to use the lim as h goes to 0 (sin h/h) as well as the identity for the tangent of the sum of two angles)"

I know that that's the derivative, but I have no idea how to prove it. Please help!

2. Write the tangent as ratio of sine over cosine, and use the definition:

$f'(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$

Note that:
$f(a+h) = \tan(a+h) = \frac{\sin(a+h)}{\cos(a+h)}$

and also:
$f(a)=\tan(a) = \frac{\sin(a)}{\cos(a)}$
The numerator becomes:
$f(a+h) - f(a) = \frac{\sin(a+h)}{\cos(a+h)} - \frac{\sin(a)}{\cos(a)}$

Put these in the common denominator to get:
$\frac{\cos(a) \sin(a+h) - \cos(a+h) \sin(a)}{\cos(a) \cos(a+h)}$