Question

Derivative for tan(x)

Answer 1

Answer:

• 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)