Understanding the Derivative of sec(x) with the Chain Rule | Step-by-Step Explanation, Examples, and Trigonometric Identity

d/dx(sec(x))

To find the derivative of sec(x) with respect to x, we can use the chain rule

To find the derivative of sec(x) with respect to x, we can use the chain rule. The chain rule states that if we have a composition of functions, in this case, sec(x), we need to take the derivative of the outer function and multiply it by the derivative of the inner function.

In this case, the outer function is sec(x), and the inner function is x.

The derivative of sec(x) is given by:

d/dx(sec(x)) = sec(x) * tan(x)

Here’s how we arrive at this result:

We start by rewriting sec(x) using its trigonometric identity:

sec(x) = 1/cos(x)

Now we can take the derivative using the quotient rule.

Let u = 1 and v = cos(x). Applying the quotient rule:

d/dx(sec(x)) = (v * d/du(u) – u * d/dv(v)) / v^2

Simplifying this expression:

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

d/dx(sec(x)) = sin(x) / cos^2(x)

Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite cos^2(x) as 1 – sin^2(x):

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

Now, using the identity of tan(x) = sin(x)/cos(x):

d/dx(sec(x)) = sec(x) * tan(x)

So, the derivative of sec(x) with respect to x is sec(x) * tan(x).

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