The derivative of tan(x) is sec2(x)

We can Proof that tan(x)' = sec2(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 + tan2(x) = sec2(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: