The derivative of the tangent of x is the secant squared of x. This is proven using the derivative of sine, the derivative of cosine and the quotient rule.
The first step in determining the tangent of x is to write it in terms of sine and cosine. The quotient identities indicate this equals the sine of x divided by the cosine of x.
The derivative of the tangent of x therefore equals the derivative of the sine of x divided by the cosine of x. This is obtained using the quotient rule. It equals the sum of the cosine squared of x plus the sine squared of x divided by the cosine squared of x.
The Pythagorean identity of sine and cosine allows this to be expressed as one divided by the cosine squared of x. It is then simplified and written as the secant squared of x.
