What is the derivative of sec x?

To find the derivative of sec x, we can use the relationship between secant and cosine. The secant function is defined as the reciprocal of the cosine function, which means that sec x = 1/cos x. This relationship will help us derive the derivative using the quotient rule or by applying the chain rule. Let's break it down step by step.

Understanding the Derivative of sec x

The derivative of a function measures how the function's output changes as its input changes. For sec x, we can express it in terms of cosine:

sec x = 1/cos x

Applying the Quotient Rule

To differentiate sec x, we can use the quotient rule, which states that if you have a function in the form of f(x) = g(x)/h(x), then the derivative f'(x) is given by:

f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2

In our case, let:

• g(x) = 1 (the numerator)
• h(x) = cos x (the denominator)

Now, we need to find g'(x) and h'(x):

• g'(x) = 0 (since the derivative of a constant is zero)
• h'(x) = -sin x (the derivative of cos x)

Calculating the Derivative

Now we can apply the quotient rule:

sec'(x) = (0 * cos x - 1 * (-sin x)) / (cos x)^2

This simplifies to:

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

Using Trigonometric Identities

We can further simplify this expression. Recall that:

tan x = sin x / cos x

sec x = 1 / cos x

Thus, we can rewrite the derivative as:

sec'(x) = sin x / (cos x)^2 = tan x * sec x

Final Result

So, the derivative of sec x is:

sec'(x) = sec x * tan x

This means that for any angle x, the rate of change of sec x with respect to x is equal to the product of sec x and tan x. This relationship is quite useful in calculus, especially when dealing with integrals and differential equations involving trigonometric functions.