Derivative of sin(3x)

Learn what is the derivative of sin (3x) with proof by first principle. Also understand how to calculate the derivative of sin(3x) by using these concepts

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of sin3x

Derivatives have a wide range of applications in almost every field of engineering and science. The sin3x differentiation can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of sin3x by applying the first principle of differentiation. In this article, you will learn what the derivative of sin 3x is and how to calculate the derivative of sin (3x) by using different approaches.

What is the derivative of sin3x?

The derivative of sin3x with respect to x is 3cos(3x), represented as d/dx(sin(3x)). This formula represents the rate of change of the trigonometric function sin 3x. In a triangle, sin x is defined as the ratio of the opposite side to the hypotenuse that makes it a fundamental concept in trigonometry. The sin 3x derivative can be calculated by using the first principle of differentiation. The derivative formula for sin(3x) can be written as;
sin x = opposite side/hypotenuse 

Differentiation of sin3x formula

The formula for the sin3x derivative is;

d/dx(sin(3x)) = 3cos(3x)

This formula provides a mathematical representation of the rate of change of the trigonometric function sin(3x) with respect to the variable x. Understanding this formula is essential in calculus and has applications in various fields such as physics, engineering, and economics. By using the formula, we can calculate the slope of the tangent line at any point on the graph of sin(3x) and analyze the behavior of the function.

How do you prove the derivative of sin3x?

There are different to derive derivatives of sin (3x). Therefore, we can prove the sin 3x differenation by using;

  1. First Principle

  2. Chain Rule

  3. Quotient Rule

Each method provids a different way to compute the sin3x differentiation. By using these methods, we can mathematically prove the formula for finding differential of sin 3x.

Derivative of sin 3x by first principle

The derivative of a function by first principle involves finding the general expression for the slope of a curve using algebra. This method is also known as the delta method. The derivative of a function measures the instantaneous rate of change, which can be defined as the limit of the difference quotient as h approaches 0. The difference quotient is given by the formula:

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

Proof of sin3x derivative by first principle

To prove the derivative of sin (3x) by using first principle, replace f(x) by sin (3x).


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


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

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

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

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

Now, by using the half-angle formula, 1- cos 3h = 2 sin2 (3h/2), the above equation is written as:

f(x) = (-sin 3x) { lim [(2 sin2 (3h/2))]/3h/2} + (cos 3x) {lim (sin 3h)/3h}

f(x) =(-sin 3x) [lim (sin(3h/2))/(3h/2). lim sin (3h/2)] + (cos 3x) {lim (sin 3h)/3h}

As we know,

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

f(x) = 0+cos 3x (3)


f(x) = 3cos (3x)

The differentiation of sin3x can also be calculated by using the derivative by definition calculator because it follows the same first principle of derivative.

Derivative of sin(3x) by chain rule

The derivative of sin3x can be calculated by using chain rule because the cosine function can be written as the combination of two functions. The chain rule of derivatives for a function y=f(g(x)) is defined as;

dy/dx = dy/du x du/dx

Where, dy/du is derivative of y=f(u) and du/dx is the derivative of u=g(x). You can also use chain rule calculator to verify the sin 3x derivative.

Proof of sin3x differentiation by chain rule

To prove the derivative of sin 3x by using the chain rule, assume that sin3x can be written as the combination of two functions. Using this let us find the sine 3x derivative.

y = sin u where u = 3x

Using chain rule,

y = cos u.du/dx


du/dx = 3

Now, using the value of u.

y = 3cos (3x)

Thus, we have derived the formula of derivative of sin (3x) by chain rule. We can also calculate the derivative of sin2x by using chain rule.

Derivative of sin 3x using quotient rule

Another method for finding the derivative of sin(3x) is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of sin3x 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 sin(3x) by quotient rule

To differentiate sin3x, we can start by writing it,

f(x) = sin (3x) = 1/ cosec (3x) =u/v

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

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

f(x) = [cosec (3x) d/dx(1) + 1. d/dx(cosec (3x))] / (cosec 3x)2

= [cosec 3x (0) - 1(-3cosec 3x cot 3x)] / cosec2 3x

= (3cosec (3x).cot (3x)) / cosec2 3x

= 3cos (3x)

Hence, we have derived the derivative of sin (3x) using the quotient rule of differentiation.

How to find the sin3x derivative with a calculator?

The easiest way to calculate the sin3x differentiation is by using an online tool. You can use our derivative differenatial calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as sin3x formula 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 sin(3x).

  2. Now, select the variable by which you want to find differentiation od sin 3x. Here you have to choose x.

  3. Select how many times you want to differentiate sin3x. In this step, you can choose 2 for second derivative, 3 for third derivative and so on.

  4. Click on the calculate button. After this step, you will get the sin 3x derivative within a few seconds.

After completing these steps, you will receive the sin3x 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 derivative of 3 sin 3x?

The derivative of 3 sin 3x is equal to the 9 cos 3x that is same as the derivative of sin (2x).

What is sin 3x formula?

The sine function is defined as the ratio of opposite sides by the hypotenuse of a triangle. The value of sin3x can be found using the trigonometric identity. sin 3x = 3sin x - 4sin3 x.

Is sin 3x the same as 3sinx?

No, sin3x is not the same as 3sin x because sin3x is the sine function value when the angle is three times x and 3 sin x is three times the value of sine

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