## 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 cos cube x can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of cos^3x by applying the first principle of differentiation. In this article, you will learn what the derivative of cos^3x is and how to calculate the differentiation of cos cube x by using different approaches.

## What is the derivative of cos cube x?

The derivative of cos^3x, denoted as d/dx(cos3x), represents the rate of change of the trigonometric function cos x with respect to the variable x. This derivative is calculated as -3cos^2x.sin x, where sin x is the sine of angle x. Cos x represents the ratio of the adjacent side to the hypotenuse of a right triangle, and is an essential component of trigonometry. By understanding the formula for the cos cube x differentiation, you can solve problems related to rates of change and optimize your understanding of calculus.

## Derivative of cos^{3} x formula

The derivative cos^3x formula is equal to the negative product of the sine and cosine function, that is;

d/dx (cos^{3} x) = -3cos^{2} x sin x

Similarly, the derivative of cos square x formula is:

d/dx (cos^{2}x)=-3cos xsin x

## How do you prove the cos^3x derivative?

There are various methods to derive derivatives of cos^{3} x. Therefore, we can prove the derivative of cos^{3} x by using;

Chain Rule

Product 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 derivative of cos^3(x).

## Derivative of cos^3x by chain rule

To calculate the cos^3x differentation, 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 derivative calculator is defined as;

dy / dx = dy / du x du / dx

## Proof of cos cube x differentiation by chain rule

To prove the derivative of cos cube x by using chain rule, we start by assuming that,

y = u^{3} where u = cos x

By the chain rule,

y = 3u^{2}. du/dx

And,

du/dx = cos x

now, using the value of u and du/dx in y, we have,

y = - 3cos^{2} x.sin x.

You can also calculate the derivative of cos^{2}x by using chain rule.

## Derivative of cos^{3} x by using product rule

The product rule is a formula used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule calculator is:

d/dx(uv) = u(dv/dx) + (du/dx)v

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

## Proof of differentiation of cos^3x by using product rule

To prove the differentiation of cos cube x by using product rule of derivatives, we have,

d/dx (cos^{3}x) = d/dx(cos^{2} x.cos x)

d/dx (cos^{3}x) = cos^{2} x. (cos x) + cos x. (cos^{2} x)

d/dx (sin^{3}x) = cos^{2} x.(- sin x)+ 2cos^{2} x(-sin x)

It can be written as;

d/dx (cos^{3}x) = -3cos^{2} x.sin x

Hence the derivative of cos cube x is proved by using the product rule.

## How to find the derivative of cos cube x with a calculator?

The easiest way to calculate the differentiation 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 different tools.

Write the function as sin^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.

Now, select the variable by which you want to differentiate cos^3x. Here you have to choose x.

Select how many times you want to differentiate cos^3x. In this step, you can choose 2 for 2nd derivative, 3 for third derivative and so on.

Click on the calculate button.

After completing these steps, you will receive the differentiation of cos cube x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

## Frequently Asked Questions

### What are the importance of derivatives?

Derivative is an important concept of calculus. It deals with the study of continuous rate of change. It plays a vital role in finance such as to keep transaction costs low in market.

### What is the difference between derivative and integral?

The derivative and integral both are important in calculus. The derivative is the rate of change of a function with respect to an independent variable. Whereas the integration is the process of finding antiderivative of a function.

### What is the derivative of cot 3x?

The derivative of cot3x is equal to the negative of 3cosec^2(3x). The derivative of cot 3x can also be calculated by using chain rule and quotient rule.