Derivative of cos(3x)

Learn what is the derivative of cos(3x). Also understand what is the formula of cosine in trigonometry and how to prove the derivative of cos 3x.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of cos 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 cos(3x) by applying the first principle of differentiation. In this article, you will learn what the differentiation of cos 3x is and how to calculate the derivative of cos3x by using different approaches.

What is the derivative of cos3x?

The derivative of cos 3x with respect to x is represented as d/dx(cos(3x)) which is equal to -3sin(3x). It is the rate of change of the trigonometric function cos(3x) and can be calculated using the first principle.

The formula for the differentiation of cos 3x is -2sin(3x), which signifies that the slope of the curve of cos (3x) decreases as x increases. In a right triangle, cos(3x) represents the ratio of the adjacent side to the hypotenuse. This relationship can be written as;
cos x = adjacent side/hypotenuse

Derivative of cos(3x) formula

The differentiation of cos3x can be calculated using the formula;

d/dx (cos(3x)) = -3sin(3x) 

Where the derivative of cos3x is equal to the negative of three times the sine function of 3x. This formula highlights that the slope of the curve of cos(3x) decreases as x increases. Additionally, the chain rule can also be applied to the formula for cos(3x) to find its derivative. Understanding the cos3x derivative is important in calculus and has various applications in mathematics.

How do you prove the derivative of cos 3x?

There are various derivative rules to derive derivatives of cos(3x). Therefore, we can prove the differentiation of cos 3x by using;

  1. First Principle

  2. Chain Rule

  3. Quotient Rule

Each method provids a different way to compute the cos(3x) differentiation. By using these methods, we can mathematically prove the formula for finding differential of cos 3x.

Derivative of cos(3x) by first principle

According to the first principle of derivative, the cos 3x derivative is equal to -3sin 3x. 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 allow us to determine the rate of change of a function at a specific point by using limit definition of derivative. This formula is also used in the limit definition calculator which is an advanced tool to calculate derivatives by using first principle.

Proof of differentiation of cos 3x by first principle

To prove the derivative of cos 3x by using the first principle, replace f(x) by cos(3x). You can also replace f(x) by cos(2x) to calculate derivative of cos(2x).

f′(x)=limh➜0f(x+h) - f(x)/h

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


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

Now, by the trigonometric formula, cos A cos B - sin A sin B = cos (A + B)

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

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

As we know,

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

f'(x) = - sin 3x (3)

Hence the derivative cos3x is equal to,

f'(x) = -3sin (3x)

Related: Also learn how the derivative of cos^2x can be calculated by using different derivative rules.

Cos3x derivative by chain rule

To calculate the derivative of cos 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 of derivatives is defined as;

dy / dx = dy / du x du / dx

Proof of derivative of cos(3x) by chain rule

To prove the derivative of cos3x by using chain rule solver, assume that cos(3x) can be written as the combination of two functions. Using this let us find the differentiation of cos 3x

y = cos u where u = 3x

Using chain rule,

y' = -sin u.du/dx


du/dx = 3

Now, using the value of u.

y' = -3sin (3x)

Thus, we have derived the formula of derivative of cos(3x) by chain rule derivative formula.

Derivative of cos(3x) using quotient rule

Another method for finding the differentiation of cos3x 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 cos3x by quotient rule

To prove the cos3x derivative, we can start by writing it,

f(x) = cos(3x) = 1/ sec (3x) =u/v

Supposing that u = 1 and v = sec (3x). Now by quotient rule calculator,

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

f(x) = [sec (3x) d/dx(1) - 1. d/dx(sec (3x))] / (sec 3x)2

= [sec 3x (0) - 1(3sec 3x tan 3x)] / sec2 3x

= (-3sec (3x).tan (3x)) / sec2 3x

= -3sin (3x)

Hence, we have derived the differentiation of cos3x using the quotient rule of differentiation.

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

The easiest way to calculate the derivative of cos 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 cos(3x)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 cos(3x).

  2. Now, select the variable by which you want to differentiate cos3x. Here you have to choose x'.

  3. Select how many times you want to differentiate cos 3x. 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 cos(3x)within a few seconds.

After completing these steps, you will receive the cos 3x derivative 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 formula for cos 3x?

The formula for cos^3x is given by cos^3x = (1/4) cos3x + (3/4) cosx. We can derive this formula using the cos3x formula.

What functions cannot be differentiable?

A function is not differentiable if its graph is a tangent line at a point. The tangent line becomes steeper as x approaches the point a until it becomes a vertical line.

Can a function have 2 derivatives?

No, a function cannot have more than one derivative. The derivative of function says that a function approaches to only one value.

Related Problems

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