Derivative for tan (x)

sec² (x)

Explanation:

Use the trigonometric ratio definition of the tangent function and the quotient rule.

tan (x) = sin (x) / cos (x)

Quotient rule: the derivative of a quotient is:

[the denominator * the derivative of the numerator less the numerator * the derivative of the denominator] / [denominator]²

(f/g) ' = [ g*f' - f*g'] / g²

So,

tan (x) ' = [ sin (x) / cos (x) ]' [ sin (x) / cos (x) ]' = [ cos (x) sin (x) ' - sin (x) cos (x) ' ] / [cos (x) ]²

= [ cos (x) cos (x) + sin (x) sin (x) ] / [ cos (x) ]²

= [ cos² (x) + sin² (x) ] / cos² (x)

= 1 / cos² (x)

= sec² (x)

The result is that the derivative of tan (x) is sec² (x)