What is the Differentiation of sec²x?

The differentiation of sec²x is a fundamental concept in calculus. Let’s break it down step by step.

First, recall that sec(x) is the reciprocal of cos(x), so sec(x) = 1/cos(x). Therefore, sec²x = (1/cos(x))² = 1/cos²x.

To differentiate sec²x, we can use the chain rule. The chain rule states that if you have a composite function f(g(x)), its derivative is f'(g(x)) * g'(x).

Let’s apply the chain rule to sec²x:

1. Let f(u) = u², where u = sec(x).
2. The derivative of f(u) with respect to u is f'(u) = 2u.
3. The derivative of u = sec(x) with respect to x is u’ = sec(x)tan(x).
4. Now, applying the chain rule, the derivative of sec²x is f'(u) * u’ = 2u * u’ = 2sec(x) * sec(x)tan(x) = 2sec²(x)tan(x).

So, the differentiation of sec²x is:

d/dx [sec²x] = 2sec²(x)tan(x)

This result is commonly used in various calculus problems, especially in integration and solving differential equations.