Derivative of sec square x

Learn what is the derivative of sec^2x with step by step examples. Also understand to prove the derivative of sec square x by chain rule and quotient rule.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of sec square x

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

What is the derivative of sec^2 x?

The sec square x derivative is equal to 2sec^2(x)tan(x) which is denoted as d/dx(sec^2x). It represents the rate of change of the trigonometric function sec^2(x).

The sec square x derivative is an important concept in calculus. The formula for the derivative is 2sec^2(x)tan(x). In simpler terms, it tells us how much the value of sec^2(x) changes as x changes. In a right triangle, sec(x) is defined as the ratio of the hypotenuse to the adjacent side, or 1/cos(x). This means that the derivative of sec^2(x) is closely related to the derivative of cos(x). By understanding the relationship between these two functions, we can gain deeper insights into the properties of trigonometric functions and their derivatives.

Derivative of sec2 x formula

The sec^2(x) derivative can be found using a simple formula. It is equal to the product of the secant and tangent functions of x. Mathematically, it is written as;
d/dx(sec^2x) = 2sec^2x tan(x)
This formula can be used to find the rate of change of sec^2(x) with respect to x. By understanding how to calculate the derivative of sec^2(x), we can solve more complex problems in calculus.

How do you prove the differentiation of sec square x?

There are different methods to derive thesec square x derivative. Two common methods are;

  1. Chain Rule
  2. Quotient Rule

Each method provides a different way to compute the secant squared derivative. By using these methods, we can mathematically prove the formula for finding the sec square x differentiation.

Derivative of sec^2 x by chain rule

To calculate the sec squared x differentiation, we can use the chain rule since the cosine function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The chain rule of derivatives is defined as;

dy / dx = dy / du x du / dx

Proof of sec^2 derivative by chain rule

To prove the differentiation of sec square x by using the chain rule, we start by assuming that,

f(x) = sec2x = 1/cos2 x = (cos x)-2

By using the chain rule of differentiation,

f(x) = (-2) (cos x)-3 d/dx (cos x)


f(x) = -2/cos3x . (-sin x)


f(x) = 2sin x/ cos2x

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

f(x) = 2sec2 x . tan x

The derivative of sec square can be easier by using the chain rule calculator with steps.

Derivative of sec squared x using quotient rule

Another method for finding the derivative of sec square is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of cosecant can also be calculated using the quotient rule. The quotient rule is defined as:

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

Proof of derivative of sec^2x by quotient rule

To prove the derivative of sec square x, we can start by writing it,

f(x)=sec2 x=1/ cos2 x =u/v

Supposing that u = 1 and v = cos2 x. Now by the quotient rule,

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

f(x) = [cos2 x d/dx(1) - 1 d/dx(cos^2 x)] / (cos^2 x)2

= [cos2 x (0) - 1 (-2cos xsin x)] / cos^4 x

= 2(cos xsin x) / cos4 2x

= 2cos xsin x)/(cos4 x)

= 2sec2 xtan x

Hence, we have derived the differentiation of sec square x using the quotient rule of differentiation.

How to find the derivative of sec2 x with a calculator?

The easiest way to calculate the derivative of secant squared is by using an online tool. You can use our derivative differentiate 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 derivative of secant squared.
  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 x. In this step, you can choose 2 for second, 3 to find 3rd derivative and so on.
  4. Click on the calculate button. After this step, you will get the differential of sec square x within a few seconds.

After completing these steps, you will receive the derivative of sec square x 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 derivative of secx?

The secx derivative is the product of sec x and tan x. It is written as;

d/dx(sec x) = sec xtan x

What is sec 2 called?

The secant squared formula is defined as the sum of the squares of the secant function and the square of the tan function. It is also known as the secant function identity square.

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