# Derivative of sin square x

Learn what is the derivative of sin square x with formula. Also understand how to prove the derivative of sin^2x by different techniques.

Alan Walker-

Published on 2023-05-26

## Introduction to the the Derivative of sin square x

Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of sin^2x can be calculated by following the rules of differentiation. Or, we can directly find the sin^2 derivative by applying the first principle of differentiation. In this article, you will learn what the sin square x derivative is and how to calculate the derivative of sin^2(x) by using different approaches.

## What is the derivative of sin^2x?

The derivative of sin^2(x) with respect to x is equal to 2sin(x)cos(x), and is denoted as d/dx(sin^2(x)). This formula represents the rate of change of the trigonometric function cos(x), which is the ratio of the adjacent side to the hypotenuse in a right triangle. In mathematical, sin(x) is defined as the ratio of the opposite side to the hypotenuse, and can be written as:

sin(x) = opposite side / hypotenuse

## Derivative of sin square x formula

The differentiation of sin square x is equal to the product of the sine and cosine functions. This can be expressed mathematically as:

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

Understanding this formula is essential in solving calculus problems related to trigonometric functions.

## How do you prove the derivative of sin^2x?

There are different ways to derive derivatives of sin2 x. Therefore, we can prove the derivative sin2 x by using;

1. Chain Rule

2. Product rule

Each method provids a different way to compute the sin square x differentiation. By using these methods, we can mathematically prove the formula for finding differential of sin^2x.

## Derivative of sin^2(x) by chain rule

To calculate the sin^2x 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 derivatives is defined as;

dy/dx = dy/du x du/dx

## Proof of derivative of sin square x by chain rule

To prove the derivative of sin 2x by using chain rule, you can use derivative chain rule calculator. Now assume that,

y = u2 where u = sin x

By chain rule,

y = 2u. du/dx

And,

du/dx = cos x

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

y = 2sinx cosx.

## Derivative of sin2 x by using product rule

The product rule is a calculus formula used to find the derivative of the product of two functions. It is expressed as;
d/dx(uv) = u(dv/dx) + v(du/dx)
Understanding and applying the product rule can be helpful in solving calculus problems that involve finding the derivatives of products of functions.

## Proof of derivative of sin^2x by using product rule

To prove the derivative of sin^2x by using product rule derivative calculator, we have,

d/dx (sin2x) = d/dx(sin x.sin x)

d/dx (sin2x) = sin x. (sin x) + sin x. (sin x)

d/dx (sin2x) = sin x.cos x+ sin x.cos x

It can be written as;

d/dx (sin2x) = 2 sin x.cos x

Hence the differentiation of sin square x is proved by using product rule.

## How to find the derivative of sin^2x with a calculator?

The easiest way to calculate the d/dx of sin^2x is by using derivatives calculator online. 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 sin^2x in the enter function box. In this step, you need to provide input value as a function as you have to calculate the sin^2 derivative.

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

3. Select how many times you want to find the differential of sin^2x. In this step, you can choose 2 for second, 3 for third order derivative and so on.

4. Click on the calculate button.

After completing these steps, you will receive the sin^2x derivative within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

### Which trigonometric functions are differentiable?

The derivative definition can be used to compute the derivatives of the elementary trig functions. Therefore, sin x and cos x is a differentiable function everywhere.

### Is slope a derivative?

The derivative measures the steepness of a function's graph at a specific point on the graph. As a result, we can see that the derivative is a slope.

### Are all functions differentiable?

All of the standard functions are differentiable except at certain singular points. For all arguments, polynomials can be differentiated. Except when q(x) = 0, where the function grows to infinity, a rational function is differentiable.