Derivative of tan 3x

Learn what is the derivative of tan (3x). Also understand the formula and ways to prove the derivative of tan 3x in trigonometry.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of tan(3x)

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of -sin x can be calculated by following the rules of differentiation. Or, we can directly find the derivative of tan3x by applying the first principle of differentiation. In this article, you will learn what the derivative of tan3x formula is and how to calculate the derivative of tan (3x) by using different approaches.

What is the derivative of tan 3x?

The tan3x derivative with respect to the variable x is equal to the square of sec x. It is denoted by d/dx(tan(3x)). The tangent, in actuality, is the slope of a line at a point of change in the function. and is defined as the ratio of sine and cosine functions in a right-angled triangle. Therefore, we can write the formula for tangent as;

tan x = sin x/ cos x

Differentiation of tan3x formula

The derivative of tan(3x) can be expressed as 3sec^2(3x) using the power rule of differentiation. This formula indicates that the derivative of tan(3x) is equal to the square of the secant function evaluated at 3x. Mathematically, it is expressed as;


The secant function is defined as the reciprocal of the cosine function and can be used to solve various trigonometric problems, including finding the length of a side of a right-angled triangle. 

How do you prove the derivative of tan(3x)?

There are different ways to derive derivatives of tan (3x). Therefore, we can prove the tan formula of derivative by using;

  1. First Principle
  2. Chain Rule
  3. Quotient Rule

Each method provides a different way to compute the tan3x derivative. By using these methods, we can mathematically prove the formula for finding differentiation of tan 3x.

How to differentiate tan3x using first principle

According to the first principle of derivative, the tan 3x derivative is equal to the square of secant function multiplied with 3. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

f(x)=lim f(x+h)-f(x) / h

This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.

Derivative of tan 3x formula proof by first principle

To prove the derivative of tan3x by using the first principle, replace f(x) by tan (3x) or you can also replace it by tan (2x) to find the derivative of tan 2x.


f(x) = lim tan 3(x+h) - tan (3x)/h


f(x) = lim [tan 3(x+h) - tan (3x)]/h

Now, by the trigonometric formula, tan x = sin x/cos x. So,

f(x) = lim [sin 3(x+h)/cos 3(x+h) - sin 3x/cos 3x]/h

f(x) = lim [sin 3(x+h) cos 3x - cos 3(x+h) sin 3x]/hcos 3(x+h) cos 3x

Since sin(x+y) = sin x cos y - cos x sin y

It is the expansion of the sin x function. Therefore, we can write the above equation as;

f(x) = lim sin 3(x+h - x)/hcos 3(x+h) cos 3x

f(x) = lim sin 3h/3h cos 3(x+h)cos 3x

More simplification,

f(x) = lim [1/cos 3(x+h)cos 3x][sin 3h/3h]

As we know,

Lim (sin 3x/3x) = 3, we get

f(x) = 3/cos 3x. cos 3x

Hence, the tan3x differentiation is verified by using first principles.

f(x) = 3sec^2(3x)

Also, you can use our derivative definition calculator to calculate derivative of tan(3x) online without doing any manual calculations. 

Derivative of tan(3x) by chain rule

To calculate the derivative of tan 3x, 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 formula of derivatives is defined as;

dy / dx = dy / du x du / dx

Proof of tan3x derivative by chain rule

To prove the differentiation of tan3x formula by using chain rule, we start by assuming that tan (3x) can be written as the combination of two functions. Using this let us find the derivative of tan3x

y = tan u where u = 3x

Using chain rule,

y = sec2 u.du/dx


du/dx = 3

Now, using the value of u.

y = 3sec2 (3x)

Thus, we have derived the formula for the derivative of tan. You can use chain rule derivative calculators to find derivatives of a function online.

Derivative of tan 3x using quotient rule

Another method for finding the derivative of sec xtan x 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 derivative of quotient formula is defined as:

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

Proof of derivative of tan(3x) by quotient rule

To prove the tan3x differentiation, we can write it,

f(x) = tan (3x) = sin (3x)/cos (3x) =u/v

Supposing that u = sin (3x) and v = cos (3x). Now by the quotient rule,

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

f(x) = [cos (3x) d/dx(sin (3x)) - sin (3x).d/dx(cos (3x))] / (cos 3xc

= [cos (3x)(3cos (3x)) + sin (3x) (3sin (3x))] / cos2 3x

= 3[cos2 (3x) + sin2 (3x)]/ cos2 3x

= 3sec2 (3x)

Hence, we have derived the derivative of tan3x using the quotient rule of differentiation.

How to find the derivative of tan(3x) with a calculator?

The easiest way to calculate the derivative of tan (3x) 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 tan(3x)in the enter function box. In this step, you need to provide input value as a function as you have to differentiate tan3x.
  2. Now, select the variable by which you want to differentiate tan3x. Here you have to choose x.
  3. Select how many times you want to differentiate tan(3x). In this step, you can choose 2 for second, 3 to find third derivative and so on.
  4. Click on the calculate button. After this step, you will get the tan3x derivative within a few seconds.

After completing these steps, you will receive the differentiation of tan3x 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 tan 2x formula?

tan2x is a double angle identity in trigonometry. The tangent function can also be written as tan2x = sin 2x/cos 2x because it is a ratio of the sine and cosine functions.

What is the tan 3x formula?

In trigonometry, the tan 3x formula is given as tan3x = (3 tan x - tan3x) / (1 - 3 tan2x). It is used to solve many mathematical problems and complex integrations. The formula for tan3x can also be written as tan3x = sin 3x/cos 3x.

What is the derivative of Secant?

The derivative of secant can be calculated by the first principle of differentiation. The derivative of secant with respect to x is sec x tan x and it is written as d/dx (sec x) = sec x.tan x.

Related Problems

Copyright © 2022 2023