What is the derivative of sec(x)?

The derivative of the secant function, sec(x), is derived using the rules of differentiation and the identity that relates secant and tangent functions. When differentiating sec(x), we apply the product rule along with the definition of secant as 1/cos(x).

Starting from the definition, the derivative of sec(x) can be computed as follows:

1. Recognize that sec(x) = 1/cos(x).

2. By using the quotient rule, we differentiate sec(x) as follows:

• The derivative of 1 is 0 (the constant).

• The derivative of 1/cos(x) involves applying the quotient rule: if u = 1 and v = cos(x), then the derivative is (v*(du/dx) - u*(dv/dx)) / v^2.

• Since du/dx = 0 (the derivative of a constant) and dv/dx = -sin(x), we substitute these into the quotient rule.

3. Simplifying the derivative yields:

• The expression results in sin(x)/cos^2(x), which can be rewritten as sec(x)tan(x) based on the definitions of tangent (tan(x) = sin(x)/cos(x