What Is the Derivative of Sec x?
The process of obtaining the derivative of a trigonometric function from a trigonometric identity, or its rate of change with respect to a variable, is known as the differentiation of trigonometric functions. The sine function’s derivative is written sin′(a)= cos(a), which means that the rate of change of sin x at a given angle x= an is determined by the cosine of that angle.
By using the quotient technique to functions like tan(x)= sin(x)/cos(x), all derivatives of circular trigonometric functions may be determined from those of sin(x) and cos(x) (x). The derivatives of the inverse trigonometric functions are obtained via implicit differentiation after these derivatives are known.
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Derivative of sec x Formula
The product of secant and tangent functions equals the differentiation or derivative of a secant function with respect to a variable. The derivative of the sec x function with respect to x equals the product of sec x, and tan x is the derivative formula.
Assuming x is a variable, the secant function is stated in trigonometric mathematical notation.
The differentiation of a sec x function with respect to x may be represented d/dx(sec x) in mathematics or as (sec x)’.
The derivative of the secant function can be expressed in terms of any variable.
- d/dh(sec h) = sec h tan h
- d/dw(sec w) = sec w tan w
- d/dy(sec y) = sec y tan y
Proof of Derivative of sec x
The product of the secant and tangent functions is the derivative of a secant function with respect to a variable. If ‘x’ is used to represent a variable, the secant function is expressed as ‘sec x’ in mathematical notation. The product of ‘sec x’ and ‘tan x’ is the differentiation of ‘sec x’ with regard to ‘x’. In differential calculus, the derivative of a secant function is determined using the first principle of differentiation.
Function Differentiation in Limit Form
First, according to the definition of the derivative, write the derivative of a function in terms of limits operation form.
If f(x) = sec x, then f(x+h) = sec(x+h).
Ways of Calculating
Quotient Rule: The Quotient Rule is a technique for obtaining a function’s derivative (differentiation) in the form of the ratio of two differentiable functions in Calculus. It is a formal rule for dividing one function by the other function in differentiation situations.
The quotient rule is based on the notion of the derivative limit. The quotient rule starts with the lowest function and concludes with the lowest function squared.
The chain rule explains how to calculate a composite function’s derivative.
It might be helpful to view the functions f and g as ‘layers of a problem.’ The chain rule advises us to differentiate the outer layer first, leaving the inner layer intact (the term f'( g(x))), and then differentiate the inner layer (the term g'(x)).
The Product Rule teaches us how to discern the difference between expressions that are the product of two simpler expressions:
Similarly, there is a derivative of tan x, a derivative of csc x, and more in the subject of trigonometry. For a better understanding, you can ask for online assistance in your homework.
Derivative of sec x: Formula, Proof by First Principle, Chain, Quotient Rule
The derivative of sec x with respect to x is equal to secx tanx. The secx is the reciprocal of cosx. In this post, we will learn how to find the derivatives of sec x using the following methods:
- First principle of derivatives
- Chain rule of derivatives
- Quotient rule of derivatives.
Table of Contents
What is the Derivative of Sec x?
The derivative of secx with respect to x is denoted by the symbol $\frac
Derivative of Sec x Formula
$(sec x) = sec x tan x, (or)
- (sec x)$’$ = sec x tan x.
Derivative of Sec x by First Principle
We will show that the differentiation of sec x is sec x tan x by the first principle. Putting f(x) = sec x in the derivative formula by first principle $f'(x)$ $=\lim\limits_
Derivative of Sec x by Chain Rule
By the chain rule, we will now show that the differentiation of sec x is sec x tan x. We have
Put t=cosx. So $\dfrac
So the derivative of secx by chain rule is secx tanx.
Derivative of Sec x by Quotient Rule
We will now find the derivative of secx by the quotient rule. This rule is used to find the derivative of a quotient function which says that
Let us express secx as a quotient function as follows.
Applying the above quotient rule of derivatives with f=1 and g=cosx, we get that
So the derivative of secx by quotient rule is secx tanx.
FAQs on Derivative of Sec x
Answer: The derivative of secx is secx tanx. This can proved using the first principle, chain rule, and quotient rule of derivatives.
Answer: As the derivative of secx is secx tanx = sinx/cos 2 x and cosx is undefined at x=0, the function secx is not differentiable at x=0.
Answer: The integration of secx is ln|secx+tanx|+C where C is an integration constant and ln is the logarithm with base e, called the natural logarithm.
Derivative of secx
Learn what is the derivative of sec x with easy and step-wise proof. Also understand to prove the derivative of secant by chain rule and quotient rule.
Published on 2023-05-26
Table of Contents
Introduction to the derivative of sec
Derivatives have a wide range of applications in almost every field of engineering and science. The secx derivative can be calculated by following the rules of differentiation. Or, we can directly find the sec x derivative by applying the first principle of differentiation. In this article, you will learn what the sec derivative is and how to calculate the derivative of sec x by using different approaches.
What is the derivative of secx?
The derivative of secant x with respect to x is given by the formula d/dx(sec x) = sec x tan x. This expression represents the rate of change of the cosine function, cos x, which is the ratio of the adjacent side to the hypotenuse in a right triangle. Specifically, sec x is the reciprocal of cos x, meaning it is equal to 1/cos x or the ratio of the hypotenuse to the base. The function sec x is represented as;
Differentiation of sec x formula
The differential of sec x can be expressed as the product of the secant and tangent functions, given by the formula d/dx(sec x) = sec x tan x. This mathematical relationship is fundamental to solving problems in calculus and trigonometry. Mathematically, it is expressed as;
How do you prove the derivative of secx?
There are different ways or methods to derive the secx derivative. These methods allow you to find the rate of change or slope of the secant function at any point, which is useful in many applications. We can prove the sec x derivative by using;
Derivative of secant by first principle
The derivative first principle tells that the derivative of sec(x) is equal to the product of sec(x) and tan(x). The derivative of a function by first principle refers to finding the slope of a curve by using algebra. It is also known as the delta method. Mathematically, the first principle of derivative formula is represented as:
This formula is also used in the limit definition of the derivative calculator, which allows you to calculate rate of change of any function online.
Proof of derivative of sec x by first principle
To prove the sec x derivative by using first principle, we start with replacing f(x) by sec x or you can replace it by sec (2x) to calculate sec2x differentiation.
Since sec x = 1/cos x, therefore,
By sum to product formulas, cos A — cos B = -2 sin (A+B)/2 sin (A-B)/2. So,
Multiply and divide by h/2,
When h approaches to zero, h/2 also approaches to zero, therefore,
We know that lim sin x/x=1,
We know that 1/cos x = sec x and sin x/cos x = tan x. So
$f'(x) = \sec x \tan x$
Hence the derivative sec is equal to the product of sec x and tan x. The above method can be also used to calculate the derivative of sec x tan x.
Differentiation of sec x by chain rule
The secx derivative proof can be found using the chain rule of derivatives, as secant x can be expressed as the composition of two functions. The chain rule of derivatives is a fundamental concept in calculus, which states that the derivative of a composite function is equal to the product of the derivatives of its component functions. Mathematically, the chain rule of derivatives is defined as;
Proof of derivative of sec x by chain rule
To prove the secx derivative by using the chain rule, we begin by assuming that,
$f(x) = \sec x = \frac<1> <\cos x>= (\cos x) -1
By using the chain rule of differentiation, we can express f'(x) as follows:
Since sin x / cos x =tan x and 1/cos x = sec x, therefore we have
Thus, we have proven that the derivative of sec x is equal to sec x times tan x. You can also use composite function calculator to easily find secx differentiation and other functions.
Differential of secx using quotient rule
The derivative of sec(x) can be also found by using the quotient rule as the secant is the reciprocal of cosine. Therefore, the differentiation of sec x can also be calculated by using the quotient rule. The quotient rule is defined as;
Proof of sec derivative by quotient rule
To prove the derivative of secant, we can start by writing it,
Supposing that u = 1 and v = cos x. Now by quotient rule,
Hence, we have derived the secx derivative using the quotient rule of derivative calculator.
How to find the derivative of sec x with a calculator?
The easiest way to calculate the sec x derivative is by using an online tool. You can use our derivative calculator with steps for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
Write the function as sec x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the secx differentiation.
Now, select the variable by which you want to differentiate sec x. Here you have to choose x.
Select how many times you want to differentiate secant x. In this step, you can choose 2 for second, 3 for third derivative and so on.
Click on the calculate button. After this step, you will get the derivative of sec within a few seconds.
After completing these steps, you will receive the secx derivative proof within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions derivative.
Frequently Asked Questions
What is the derivative of 2 sec x?
The derivative of 2 sec x is similar to the secx differentiation. It is 2 secx . tanx. It is written as;
What is secx equal to?
The secant is equal to the ratio of 1 and cosine. Since the cosine is the ratio between base and hypotenuse of a triangle. Therefore, secx will be the reciprocal of this ratio.
What is the derivative in basic calculus?
In mathematics, the derivative is the rate of change of a function with respect to a variable. Derivatives are essential for solving calculus and differential equation problems.
Что за функция sec?
СЕКАНС, в ТРИГОНОМЕТРИИ — отношение длины гипотенузы к длине катета, прилежащего к данному острому углу в прямоугольном ТРЕУГОЛЬНИКЕ. Секанс угла А обычно сокращенно записывается, как sec А, и равен обратному КОСИНУСУКОСИНУСУКОСИНУС — КОСИНУС, а, муж. (спец.). Тригонометрическая функция угла, в прямоугольном треугольнике равная отношению к гипотенузе катета, прилежащего к данному острому углу. https://dic.academic.ru › dic.nsf › ushakov
Что такое sec в тригонометрии?
Секанс ( sec ) (\sec) (sec) Секанс — это отношение, обратное косинусу. Это отношение гипотенузы к прилежащему к углу катету в прямоугольном треугольнике.
Что такое sec U?
ЭСУД / МПСЗ SECU-3 это комплексный проект, который включает в себя множество средств и решений для тюнинга двигателя внутреннего сгорания (ДВС), от простой микропроцессорной системы зажигания (МПСЗ) и до полнофункциональной электронной системы управления двигателем (ЭСУД).
Что такое sec в Photomath?
Функция SEC — это одна из математических и тригонометрических функций. Она возвращает секанс заданного угла в радианах. где x — это угол в радианах, секанс которого требуется вычислить.