A function is an expression, or a rule, or some law in mathematics that defines a relationship between an independent and a dependent variable.

The derivative of a function of a real variable quantifies the sensitivity of the function value (output value) to a change in its argument (input value). Calculus relies heavily on derivatives. For example, the velocity of a moving object is the derivative of its position with respect to time: it measures how quickly the object’s position changes as time passes.

When a derivative of a function of a single variable lies at some specific input value, it is the slope of the tangent line to the function’s graph at that particular point. The tangent line is the function’s best linear approximate value near that input value. As a result, the derivative is frequently known as the **instantaneous rate of change. **However, it is also called the ratio of the instantaneous change of a dependent variable to that of the independent variable.

The derivative of tan x, sec x & tan x is very important in calculus.

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**Derivative of tan x**

The derivative of tan x, sec x & tan x is very important in calculus.

sec^{2}x is the derivative of tan x.

You will need to know a few results to understand why. To begin, you must understand that the derivative of sin x is cos x. Here’s an example of a first-principles result.

Once you know this, you can deduce that the derivative of cos x is sin x (which you will need later). You should also be aware of the Quotient Rule for differentiation:

When all of those pieces are in place, the differentiation is as follows:

**Domain and range of tan x**

Just like the derivative of tan x, sec x & tan x is very important in calculus, it is also important to know the range and domain. This is because the length of the base in a right triangle is 0 and **cos x** **= 0** when (x = k/2), where k is an odd integer, the tangent function is not defined at odd multiples of π/2. As a result, the domain of tan x includes all real numbers except odd multiples of π/2. So, as the value of tan x varies from negative infinity to positive infinity, the range of the tangent function now includes all real numbers. Thus, as a result, we can conclude:

- R – ((2k+1)π/2), where k is an integer
- Range is equal to R, where R is the set of real numbers.

**Properties of tan x**

We know that, the derivative of tan x, sec x & tan x is very important in calculus. Let’s go over some of the tangent function’s most important properties. The basic properties of tan x, as well as its value at various angles and the trigonometric identities involving tan x, are as follows:

- The tangent function is an odd function —- [tan (-x) = -tan x].
- Tan x is not defined at values of x where cos x = 0.
- The graph of tan x contains an infinite number of vertical asymptotes.
- The values of the tangent function at particular angles are:
- tan 0 = 0
- tan π/6 = 1/√3
- tan π/4 = 1
- tan π/3 = √3
- tan π/2 = Not defined

- The trigonometric identities are as follows:
- 1 + tan
^{2}x = sec^{2}x - tan 2x = 2 tan x/(1 – tan
^{2}x) - tan (a – b) = (tan a – tan b)/(1 + tan a tan b)
- tan (a + b) = (tan a + tan b)/(1 – tan a tan b)
- The graph of tan x is symmetric with respect to the origin.
- The x-intercepts of tan x are where sin x takes the value zero, that is, when x = nπ, where n is an integer.

**Period of tan x **

We have already seen the graph of tan x. So, there,

tan x= tan(x+π) = tan(x+2π) = Tan(x+3π)..

So, fundamental period of tan x is π.

Again, in other words tan x is a periodic function of a period π.

**Derivative of sec x tan x**

The derivative of tan x, sec x & tan x is very important for you to learn in calculus.

Firstly, Assume we have y=f(x).g(x)

Then, applying the Product Rule, y’=f(x).g'(x)+f'(x)g(x)

So, in simple terms, keep the first term the same and differentiate the second term, then differentiate the first term and keep the second term the same, and so on.

**Derivative of tan root x**

**Derivative of tan x^-1**

The inverse of x is the same as the inverse of the tan function. So, in other words,

if y = tan x, then x = tan^{-1}(y). So, tan^{-1} is the inverse function of tan in this case.

Now, we can find the derivative of tan^{-1}x in the following way:

=

**Derivative of tan x^2**

When we have a composite function in differential calculus, we use the Chain Rule. Thus, the derivative will be equal to the derivative of the outside function with respect to the inside function multiplied by the derivative of the inside function. So, now, let’s take a look at how that works mathematically.

The Chain Rule

f‘(g(x))⋅g‘(x)

Firstly, assume we have the composite function sin(4x). We are aware of:

f(x)= sin x

⇒f‘(x)= cos x

g(x)= 4x

⇒g‘(x)=4

So, as a result, the derivative will be equal to

cos(4x)⋅4

=4cos(4x)

Thus, all we have to do now is find our two functions, find their derivatives, and enter them into the Chain Rule expression. So, using the chain rule for derivative of tan x^2

Hence, using the chain rule,

But, remember that, if

**What is the derivative of sec2x**

**What is sec x**

Cosecant, secant, and cotangent [cosec, sec, cot] are the secondary trig functions. They are ratios that relate opposite, adjacent, and hypotenuse side lengths to angles in a right triangle. So sec x is simply the ratio of a hypotenuse’s length to the length of an adjacent side. Read Also: What is the derivative of tangent ?

**What are the 6 trig functions’ derivatives?**

Trigonometric Function Derivatives — The six basic trigonometric functions are sine (sin x), cosine (cos x), tangent (tan x), cotangent (cot x), secant (sec x), and cosecant (cosec x). Again, in their respective domains, all of these functions are continuous and differentiable.

**What is the differentiation of theta?**

It is determined by the derivative of the variable with respect to the variable. For instance, if you took the derivative of with respect to, you would get one. However, because x is the most commonly used variable, the derivative is usually taken with respect to x.

**tan x graph**

The tangent function graph is discontinuous because the value of tan x is not defined at odd multiples of /2, that is, tan x is not defined for x = k/2, where k is an odd integer. Furthermore, because the tangent function has a period of its values repeat every radian, and thus the pattern of the curve is repeated every radian. Additionally, as shown in the graph of the tangent function below, the function has vertical asymptotes at x = -/2, /2, -3/2, 3/2, 5/2, and so on. The graph of tan x is given as follows:

**sec x tan x graph**

The graph of sec x tan x is given as follows :

**Is tan x available indefinitely?**

The function tan x is not continuous, but it is continuous on some intervals, such as the interval /2, which has infinite discontinuities.

**How does one go about locating a derivative?**

Using the limit definition of derivatives, we can compute the derivative of f(x) by performing the following steps:

- Determine f(x + h).
- In the limit definition of a derivative, enter f(x + h), f(x), and h.
- Reduce the difference quotient’s complexity.
- Take the simplified difference’s limit as h approaches zero.

**What is ‘cot x’?**

cot is an abbreviation for ‘cotangent.’ This is the reciprocal of the trigonometric function ‘tangent,’ also known as tan (x). As a result, cot(x) can be reduced to 1/tan (x). An alternative way to write 1/tan(x) using trigonometric rules is cos(x)/sin(x) (x).

**Derivative of tan x, sec x & tan x FAQs**

**1. ****What exactly is a derivative in calculus?**

Ans. The derivative of tan x, sec x & tan x is very important in calculus. Thus, the derivative is defined as the slope of a line that is tangent to the curve at a specific point. Again, an alternate derivative definition is the limit of the function’s instantaneous rate of change as the time between measurements approaches zero. For example, the derivative of tan x, sec x & tan x as mentioned in the article above.

**2. ****What exactly is tan x?**

Ans. The tangent function, one of the six basic trigonometric functions, is commonly written as tan x. In a right-angled triangle, it is the ratio of the opposite and adjacent sides of the angle under consideration. Additionally, there are several trigonometric identities and formulas related to the tangent function that can be derived using various formulas. Period = π/|b| is the formula for the period of the tangent function f(x) = a tan (bx). Because b = 1 in tan x, the tangent function tan x is a periodic function with a period of π/1 =.π.

**3. ****Why do you find the derivative?**

Ans. The derivative of a function at a given point, as we have seen, gives us the rate of change or slope of the tangent line to the function at that point. Thus, we can get the velocity at a given time by differentiating a position function at that time. So, the original purpose of the derivative is to examine a function’s sensitivity or rate of change with respect to its independent variable — that is, how much does the dependent variable, y, respond to a small change in the independent variable, x?

For example, The derivative of tan x, sec x & tan x is very important for you to learn in calculus.

**4. ****What exactly is sec x?**

Ans. In a right-angled triangle, if x is the angle in between base and hypotenuse. sec x defines the ratio of the length of the hypotenuse to that of the base.

**5. ****What is the difference between derivative and differentiation?**

Ans. In terms of interrelationship, the terms differential and derivative are inextricably linked. So, in mathematics, variables are changing entities, and the rate of change of one variable with respect to another is generally a derivative.

Again, differential equations are equations that define the relationship between these variables and their derivatives. The process of determining a derivative is known as differentiation. A function’s derivative is the rate of change of its output value with respect to its input value, whereas its differential is the actual change of the function.

**6. ****How do you know when to use tan?**

Ans. You can remember the phrase “SOME PEOPLE HAVE CURLY GRAY HAIR TURNED PERMANENTLY BLACK” to remember the formulas.

So, Use sine if you have the hypotenuse and the opposite side.

Again, use cosine if you have the hypotenuse and the adjacent side.

Again, you can use tangent when you have adjacent and opposite sides.

Alternatively, if you have any two numbers from the first group (say, an angle and the hypotenuse and you need to find the length of the opposite side), then use that group. The same goes for the other two groups.