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;

$\sec x =\frac{1}{\cos x} = \frac{hypotenuse}{base}$

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;

$\frac{d}{dx}(\sec x) = \sec x\tan x$

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;

1. First Principle

2. Chain Rule

3. Quotient Rule

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:

$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$

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.

$f'(x) = \lim_{h\to 0} \frac{\sec(x+h)-\sec x}{h}$

Since sec x = 1/cos x, therefore,

$f'(x) = \lim_{h\to 0}\frac{{\frac{1}{\cos(x+h)} -\frac{1}{\cos x}}{h}$

More simplifying,

$f(x) = \lim_{h\to 0} \frac{\cos x -\cos (x+h)}{h\cos(x+h)\cos x$

By sum to product formulas, cos A - cos B = -2 sin (A+B)/2 sin (A-B)/2. So,

$f'(x) = \frac{1}{\cos x}\lim_{h\to 0}\frac{1}{h}\frac{- 2\sin \frac{x + x + h}{2}\sin \frac{x - x - h}{2}}{\cos(x + h)}$

$f'(x)= \frac{1}{\cos x}\lim_{h\to 0}\frac{1}{h} \frac{-2\sin\frac{2x + h}{2}\sin\frac{- h}{2}}{\cos(x + h)}$

Multiply and divide by h/2,

$f'(x)= \frac{1}{\cos x}\lim_{h\to 0}(\frac{1}{h})(\frac{h}{2})\frac{-2 \sin \frac{2x + h}{2}\frac{\sin \frac{-h}{2}}{h/2}}{\cos(x + h)}$

When h approaches to zero, h/2 also approaches to zero, therefore,

$f'(x) = \frac{1}{\cos x}\lim_{h\to 0}\frac{\sin (h/2)}{h/2} \lim_{h\to 0}\frac{\sin\frac{2x + h}{2}}{\cos(x + h)}$

We know that lim sin x/x=1,

$f'(x) = \frac{1}{\cos x}.1. \frac{\sin x}{\cos x}$

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;

$\frac{dy}{dx} = \frac{dy}{du}\times \frac{du}{dx}$

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:

$f'(x) = (-1) (\cos x)^-2 \frac{d}{dx} (\cos x)$

Simplifying,

$f'(x) = -\frac{1}{\cos²x}\cdot (-\sin x)$

Again,

$f'(x) = \frac{\sin x}{\cos²x}$

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

$f'(x) = \sec x.\tan x$

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;

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f(x).g'(x)-g(x).f'(x)}{(g(x))^2}$

Proof of sec derivative by quotient rule

To prove the derivative of secant, we can start by writing it,

$f(x)=\sec x=\frac{1}{\cos x} =\frac{u}{v}$

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

$\frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u(x).v'(x)-v(x).u'(x)}{(v(x))^2}$

$f'(x) = \frac{\cos x \frac{d}{dx}(1) - 1 \frac{d}{dx}(\cos x)}{(\cos x)^2}$

$f'(x)= \frac{\cos x (0) - 1 (-\sin x)}{(\cos^2x)}$

$f'(x)= \frac{\sin x}{\cos^2x}$

$f'(x)= \frac{1}{\cos x}\times\frac{\sin x}{\cos x}$

$f'(x)=\sec x\tan x$

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.

1. 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.

2. Now, select the variable by which you want to differentiate sec x. 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 for third derivative and so on.

4. 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;

$\frac{d}{dx}(2\sec x) = 2\sec x\tan x$

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.