# Derivative of Sin Cube x

Learn what is the derivative of sin cube x with step-by-step proof. Also understand how to solve derivative of sin^3x.

Alan Walker-

Published on 2023-05-26

## Introduction to the differentiation of sin cube x

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

## What is the derivative of sin^3x?

The derivative of sin cube x with respect to x can be expressed as d/dx(sin^3(x)) = 3sin^2(x)cos(x). This formula represents the rate of change of the trigonometric function cos(x), which relates to the ratio of the opposite side to the hypotenuse in a triangle. By understanding the derivative of sin cubed, you can more easily calculate the rate of change of various functions that involve this trigonometric expression.

## Derivative of sin3 x formula

The differentiation of sin^3x can be expressed as the product of the sine and cosine functions. Specifically, the formula for the derivative of sin^3(x) is;

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

This formula is commonly used in calculus and other fields of mathematics and science to calculate the rate of change of trigonometric functions involving sin^3(x).

## How do you prove the derivative of sin^3x?

There are different ways to derive derivative of sin cube x. Therefore, we can prove sin cube x differentiation by using;

1. Chain Rule

2. Product rule

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

## Derivative of sin^3x by chain rule

To calculate the sin^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 is defined as;

dy/dx = dy/du x du/dx

## Proof of differentiation of sin cube x by chain rule

To prove the derivative of sin^3x by using chain rule, we start by assuming that,

y = u3 where u = sin x

By chain rule,

y = 3u2. du/dx

And,

du/dx = cos x

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

y = 3sin2 x cosx.

Also, you can use our chain rule finder to calculate derivative of a combination of two functions.

## Derivative of sin3 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 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 derivative of sin^3x by using product rule

To prove the differentiation of sin cube x by using product rule of derivative, we have,

d/dx (sin3x) = d/dx(sin2 x.sin x)

d/dx (sin3x) = sin2 x. (sin x) + sin x. (sin2 x)

d/dx (sin3x) = sin2 x.cos x.+ 2sin2 x.cos x

It can be written as;

d/dx (sin3x) = 3sin2 x.cos x

Thus, we have proved the derivative of sin^3(x) using the product rule of differentiation calculator.

## How to find the derivative of sin^3x with a calculator?

The easiest way to calculate the derivative of sin cube x 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 derivative calculator with steps.

1. 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 sin^3x.

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

3. Select how many times you want to differentiate sin^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 sin^3x within a few seconds.

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

### What are derivatives in simple words?

The derivative is known as the rate of change in a quantity with respect to another quantity. In mathematics, the derivative is the instantaneous rate of change in a function with respect to an independent variable.

### Why do we use derivatives?

The derivative is used to calculate the rate of change of a quantity with respect to another quantity. Moreover, the equation of tangent and normal line to a curve of a function can also be calculated by using derivatives.

### Why is the derivative of sinx is cosx?

The derivative of sin x is cos x because cos x=0 at all the points at which sin x has either maximum or minimum value. It means that at all critical points of sin x, the cos x is zero. Therefore the derivative of sin x is cos x.