**The derivative of tan(x)**** is ****sec**^{2}(x)

We can Proof that tan(x)' = sec^{2}(x) from the derivatives of sine and cosine.

Given: sin(x)' = cos(x); cos(x)' = -sin(x);

We know that tan(x) =^{ sin(x) }/_{ cos(x).}

tan(x)' = (^{sin(x)}/_{cos(x)})'

Now, using the Quotient Rule ^{(*)}

= ^{( cos(x)sin(x)' - sin(x)'cos(x) ) }/ _{cos2(x) }

= ^{( cos(x)cos(x) + sin(x)sin(x) )} / _{cos2(x) }

And finally:

**tan(x) = 1 + tan**^{2}(x) = sec^{2}(x)

^{(*)}** The quotient rule**

In Calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.

If the function one wishes to differentiate, f(x), can be written as

**f(x) =**^{ g(x)}/_{h(x})

and h(x) not iqual to zero, then the rule states that the derivative of ^{g(x)}/_{h(x})

**f'(x) = (g'(x)h(x) - h'(x)g(x)) / [h(x)]**^{2}

References:

http://www.math.com/tables/derivatives/more/trig.htm

https://en.wikipedia.org/wiki/Quotient_rule