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Home / Mathematics / Derivative of sec x

SAT Derivative of sec x: Definition, formula, proof and examples

Last Updated on Apr 01, 2025
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Derivative of sec x

The ratio of hypotenuse to the adjacent side of an angle in a right triangle is defined as the trigonometric function secant of an angle.

where A denotes the angle, c the hypotenuse, and b the adjacent side

One of the first transcendental functions introduced in Differential Calculus is the Secant Derivative. Secant times tangent, or \sec x .\tan x is the derivative of the secant function (x). where A denotes the angle, c the hypotenuse, and b the adjacent side

This derivative can be proven using limits and trigonometric identities.

\frac{d}{dx}\left ( \sec x \right )

\left ( \sec x \right )’ =\sec x .\tan x

Derivative of Sec X Using First Principle

We will use first principles (or the definition of the derivative to demonstrate that the derivative of sec x is sec x tan x. When attempting to find the derivative, avoid using the first principle unless specifically mentioned in the question. Using the first principle may take longer than the traditional method and may necessitate a good understanding of various forms of formulas related to algebra, trigonometry, and a bit of manipulation. Let’s start.

Assume that

Hence, proved.

Derivative of Sec X Using Quotient Rule

Using the quotient rule, we will demonstrate that the differentiation of sec x with respect to x yields sec x tan x.

We will assume that f\left ( x \right )= \sec x, which can be written as f\left ( x \right )= \frac{1}{\cos x}

Hence proved

Derivative of Sec x by Chain Rule

It is ,

\frac{\mathrm{d} }{\mathrm{d} x}\frac{1}{\cos \left ( x \right )} =\frac{\mathrm{d} }{\mathrm{d} x} \left ( \cos \left ( x \right )^{-1} \right ) \)

This is equivalent to the chain rule.

Hence proved

Solved Examples of Derivative of sec x

Problem: 1What is derivative?

Solution:

By using power rule and chain rule

 

The derivative of given function is

Problem: 2Determine the derivative of

Solution:

Then,

…..(1)

Differentiating both sides in terms of x,

According to one of the trigonometric identities,

Derivative of is

Grasping the derivative of sec x is essential for proficiency in calculus, particularly for the solution of problems relating to rates of change and integrals. Using the rules such as the quotient rule and chain rule, we deduced that d/dx (sec x) = sec x tan x. The result comes into play when performing advanced math involving physics and engineering applications. With a firm understanding of this differentiation process, you can comfortably challenge associated trigonometric derivatives and use them for practical problems, strengthening your total calculus abilities.

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Derivative of Sec X FAQs

By using below formula we can find the derivative of sec x

We will use first principle to demonstrate that the derivative of is

No, it is not.

Integration by parts cannot be used to calculate the integration of a secant function. As a result, finding its integral necessitates a unique procedure.

Yes, it is. is defined as the ratio of two continuous functions.
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