what is the derivative of sec^2x

what is the derivative of sec^2x

The Correct Answer and Explanation is:

To differentiate the function sec⁡2(x)\sec^2(x)sec2(x), we can use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

We start by recognizing that sec⁡2(x)\sec^2(x)sec2(x) is a composition of two functions: sec⁡(x)\sec(x)sec(x) and the exponentiation function. Specifically, we can write:sec⁡2(x)=[sec⁡(x)]2\sec^2(x) = [\sec(x)]^2sec2(x)=[sec(x)]2

Now, we need to apply the chain rule. First, differentiate the outer function, which is u2u^2u2 where u=sec⁡(x)u = \sec(x)u=sec(x). The derivative of u2u^2u2 with respect to uuu is 2u2u2u. Then, we multiply by the derivative of the inner function, which is sec⁡(x)\sec(x)sec(x). The derivative of sec⁡(x)\sec(x)sec(x) is sec⁡(x)tan⁡(x)\sec(x) \tan(x)sec(x)tan(x). Therefore, the derivative of sec⁡2(x)\sec^2(x)sec2(x) is:ddx(sec⁡2(x))=2sec⁡(x)⋅sec⁡(x)tan⁡(x)\frac{d}{dx} \left( \sec^2(x) \right) = 2 \sec(x) \cdot \sec(x) \tan(x)dxd​(sec2(x))=2sec(x)⋅sec(x)tan(x)

This simplifies to:2sec⁡2(x)tan⁡(x)2 \sec^2(x) \tan(x)2sec2(x)tan(x)

Thus, the derivative of sec⁡2(x)\sec^2(x)sec2(x) is 2sec⁡2(x)tan⁡(x)2 \sec^2(x) \tan(x)2sec2(x)tan(x).

Explanation:

1. Secant and Tangent Relationship: The secant function is the reciprocal of cosine, i.e., sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}sec(x)=cos(x)1​. The tangent function is the ratio of sine to cosine, i.e., tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)​. These trigonometric identities often appear when working with derivatives of secant and other related functions.
2. Chain Rule: Since sec⁡2(x)\sec^2(x)sec2(x) is a composition of functions, we use the chain rule to differentiate. The chain rule allows us to differentiate complex expressions by breaking them down into simpler parts, which is essential when dealing with functions like sec⁡2(x)\sec^2(x)sec2(x) where one function is nested inside another.
3. Simplification: After applying the chain rule, we simplify the result to 2sec⁡2(x)tan⁡(x)2 \sec^2(x) \tan(x)2sec2(x)tan(x), which is the final answer.

This method is fundamental in calculus and is commonly used when working with trigonometric functions and their derivatives.