Derivative of sec(x) - Proof and Explanation

To find the derivative of $\sec (x)$, we start by rewriting it in a form that might be easier to differentiate. Recall that:

To differentiate this, we use the <strong>quotient rule</strong>. The quotient rule states that for two functions $u(x)$ and $v(x)$, the derivative of their quotient $\frac{u}{v}$ is given by:

In our case:

• $u=1$ (the numerator)
• $v=\cos (x)$ (the denominator)

Step 1: Differentiate $u$ and $v$

1. The derivative of $u$ with respect to $x$ is $0$ because $u=1$ – a constant.
2. The derivative of $v=\cos (x)$ with respect to $x$ is $-\sin (x)$.

Step 2: Apply the quotient rule

Plugging these into the quotient rule:

This simplifies to:

Step 3: Simplify the result

Now, we can rewrite this expression in terms of other trigonometric functions:

Thus, the derivative of $\sec (x)$ is:

Q.E.D.