The Derivative of the Secant Function | How to Find and Simplify

Derivative of sec x

To find the derivative of the secant function, we can use the quotient rule

To find the derivative of the secant function, we can use the quotient rule.

The secant function is defined as: sec(x) = 1/cos(x), where cos(x) is the cosine function.

Using the quotient rule, the derivative of sec(x) is given by:

d/dx(sec(x)) = (cos(x)d/dx(1) – 1/cos^2(x)d/dx(cos(x)))/(cos(x))^2

Now, let’s evaluate each term separately:

d/dx(1) = 0, since the derivative of a constant is always zero.

d/dx(cos(x)) = -sin(x), which is the derivative of the cosine function.

Substituting these values back into our equation, we get:

d/dx(sec(x)) = (0 – 1/cos^2(x)(-sin(x)))/(cos(x))^2

= sin(x)/cos^2(x)

Now, we can simplify this expression further. Using the trigonometric identity:

sin(x) = 1 – cos^2(x),

we can rewrite sin(x) as (1 – cos^2(x)). Thus, we have:

d/dx(sec(x)) = [(1 – cos^2(x))/cos^2(x)]/(cos(x))^2

= (1 – cos^2(x))/(cos^2(x) * cos^2(x))

= 1/(cos^2(x))

= sec^2(x)

Therefore, the derivative of sec(x) is equal to sec^2(x).

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