Derivative of sec 2x

Learn what is the derivative of sec square x with formula. Also understand how to prove the derivative of sec 2x by three techniques.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of sec 2x

Derivatives have a wide range of applications in almost every field of engineering and science. The sec 2x derivative can be calculated by following the rules of differentiation. Or, we can directly find the sec2x differentiation by applying the first principle of differentiation. In this article, you will learn what the sec 2 x derivative is and how to calculate the sec2x derivative by using different approaches.

What is the derivative of sec(2x)?

The derivative of sec(2x) with respect to x is 2sec(2x)tan(2x), represented by d/dx(sec(2x)). This derivative measures the rate of change of the cos(2x) function. In a right-angled triangle, sec(x) is the reciprocal of cos(x), which represents the ratio of the hypotenuse to the adjacent side. Thus, sec(x) can be written as;

sec x=hypotenuse/adjacent or 1/cos(x)

By understanding the sec 2x derivative,  you can solve problems involving trigonometric functions and their rates of change.

Derivative of sec2x formula

The differentiation formula for sec 2x is the product of the secant and tangent functions. This formula can be written as:

d / dx (sec 2x) = 2sec 2xtan 2x

By using this formula, you can find the derivative of sec(2x) with respect to the variable 'x'. The result is equal to 2 times the secant of 2x multiplied by the tangent of 2x. This formula is useful for solving problems involving trigonometric functions and their derivatives.

How do you prove the sec(2x) derivative?

There are various ways to prove the differentiation of sec 2x. These are;

  1. Chain Rule

  2. Quotient Rule

  3. Product rule

Each method provids a different way to compute the sec2x differentiation. By using these methods, we can mathematically prove the formula for finding differential of cos x.

Derivative of sec 2x by chain rule

The derivative of sec2x can be calculated using the chain rule of differentiation, which applies when the secant function can be expressed as a combination of two functions. 

dy/dx = dy/du x du/dx

In this formula, y is a function of u, which is a function of x. The chain rule allows you to find the derivative of the composite function y(u(x)), which is written as dy/dx. By breaking down the composite function into its component parts and applying the chain rule, you can calculate the derivative of sec(2x) with respect to x.

Proof of differentiation of sec2x by chain rule

To prove the sec2x derivative by using chain rule, we start by assuming that,

f(x) = sec 2x = 1 / cos 2x = (cos 2x)-1

By using chain rule of differentiation,

f'(x) = (- 1) (cos x) - 2 d / dx (cos x)


f'(x) = - 2 /(cos 2x)2.(-sin 2x)


f'(x) = 2sin 2x/ (cos 2x)2

Since sin x / cos x = tan x and 1 / cos x = sec x, therefore we have

f'(x) = 2sec 2x.tan 2x

Thus, we have proved the sec2x derivative. You can also use multivariable chain rule calculator to help with this calculation.

Derivative of sec 2x using quotient rule

Another method for finding the sec2x differentiation is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The quotient rule of two functions is defined as:

d / dx (f/g) = f(x). g'(x) - g(x).f'(x) /{g(x)}2

Proof of derivative of sec2x by quotient rule

To prove the sec2x derivative, we start by writing it as,

f(x) = sec 2x = 1 / cos 2x = u/v

Supposing that u = 1 and v = cos 2x. Now by quotient rule,

f'(x) = (vu' - uv')/v2

f'(x) = [cos 2x d / dx(1) - 1. d / dx(cos 2x)] / (cos 2x)2

f'(x)= [cos 2x(0) - 1 ( - 2sin 2x)] / cos22x

f'(x)= 2(sin 2x) / cos22x

f'(x)= 2/cos 2x . (sin 2x)/(cos 2x)

f'(x)= 2sec 2x.tan 2x

Hence, we have derived the sec2x derivative. You can also use the quotient calculator.

Differentiation of sec 2x by using product rule

The product rule in derivatives is used when we have to calculate derivative of two functions at a time. The product rule is;

[uv]' = u.v' + u'.v

This rule is useful in solving problems involving the rates of change of quantities that are dependent on two or more variables.

Proof of derivative of sec 2 x by using product rule

To differentiate sec 2x by using product rule of derivatives, we have,

d / dx (sec 2x) = d / dx(1.sec 2x)

d / dx (sec 2x) = 1. (sec 2x)' + sec 2x. (1)'

d / dx (sec 2x) =2sec 2x.tan 2x + sec x.(0)

It can be written as;

d / dx (sec 2x) = 2sec 2x.tan 2x


d / dx (sec 2x) = 2sec 2x.tan 2x

Hence the differentiation of sec2x is proved by using product rule. Also verify the derivative of sec by using product rule.

How to find the sec2x differentiation with a calculator?

The easiest way to differentiate sec2x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step - by - step way to calculate derivatives by using this tool.

  1. Write the function as sec 2x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the sec(2x) derivative.

  2. Now, select the variable by which you want to differentiate sec 2x. Here you have to choose 'x'.

  3. Select how many times you want to differentiate secant 2x. In this step, you can choose 2 for second, 3 for the third derivative and so on.

  4. Click on the calculate button. After this step, you will get the ddifferentiation of sec2x within a few seconds.

After completing these steps, you will receive the derivative of sec 2x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

Frequently Asked Questions

What is the formula of sec 2x?

The function secant is a trigonometric function that is the reciprocal of opposite to hypotenuse of a triangle. It can be written by using tangent function. Such as;

sec2x = 1 + tan2x

What is the derivative of tangent?

The differentiation of tanx is the square of secant function and the derivative of cotangent is the square of cosecant function.

Is sec the inverse of cos?

Yes, secant is the inverse of cosine function. Or in other words, it can be written as the reciprocal of cosine function as;

sec x = 1 / cos x

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