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


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.

Frequently Asked Questions

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.

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