Introduction to the Derivative of sin x

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

What is the derivative of sinx?

The derivative of sin(x), represented as d / dx(sin x), is equal to cos x. It represents the rate of change of the trigonometric function sin x, which is defined as the ratio of the opposite side to the hypotenuse in a triangle.

By understanding the sinx derivative, we can calculate how quickly the sine function is changing at any given point. This concept is not only important in mathematics, but also in many other fields such as physics and engineering.

Derivative sin x formula

The formula for finding the differentiation of sin x is simple and easy to remember: it is equal to the cosine function. This formula can be expressed mathematically as;

d/dx(sin x) = cos x

By using this formula, we can quickly and accurately calculate the derivative of sinx at any given point. 

Related: Also learn how to find the derivative of sin square x.

How do you prove the derivative of sin x?

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

1. First Principle

2. Chain Rule

3. Quotient Rule

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

Derivative of sin x by first principle

According to the first principle of derivative, the sin x derivative is equal to cos x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

f(x)=lim_h→0 f(x+h)-f(x) / h

This formula allow us to determine the rate of change of a function at a specific point by finding derivative using definition calculator.

Proof of derivative of sinx by first principle

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

f'(x)=lim_h→0f(x+h) - f(x) / h

f'(x) = lim_h→0 sin (x+h) + sin x / h

Therefore,

f'(x) = lim_h→0 [sin(x+h) + sin x] / h

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

f'(x) = lim_h→0 [sin x cos h + cos x sin h - sin x] / h

f'(x) = lim_h→0 [-sin x(1 - cos h) + cos x sin h] / h

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

f'(x) = (-sin x) { lim_h→0 [(2 sin² (h / 2))] / h} + (cos x) {lim_h→0 (sin h) / h}

f'(x) =(-sin x) [lim_h→0 (sin(h / 2)) / (h / 2). lim_h→0 sin (h / 2)] + (cos x) {lim_h→0 (sin h) / h}

As we know,

lim_h→0 (sin x / x) = 1, we get

f'(x) =- sin x (1. sin (0 / 2)) + cos x (1)

f'(x) = -sin x(0) + cos x

f'(x) =  cos x

Differentiation of sinx by chain rule

To calculate the sin x 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 x by chain rule

To prove the derivative sin x by using chain rule, assume that sin x can be written as sin x = cos (π / 2 + x). Using this let us find the derivative of sinx.

Using chain rule,

dy / dx = sin(π / 2 + x) . d / dx (π / 2 + x)

dy / dx = sin(π / 2 + x) . (1)

dy / dx = sin(π / 2 + x)

dy / dx = cos x

Thus, we have derived the formula of differential sin x by chain rule formula calculator.

Sin derivative using quotient rule

Another method for finding the differential of -sin x is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The derivative 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 x by quotient rule

To prove the derivative sin x, we can write it,

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

Supposing that u = 1 and v = cosec x. Now by quotient rule,

f'(x) = (vu - uv) / v²

f'(x) = [cosec x d / dx(1) - d / dx(cosec x)] / (cosec x)²

f'(x) = [cosec x (0) - 1 (-cosec x cot x)] / cosec²x

f'(x) = (cosec x.cot x) / cosec²x

f'(x) = cos x

Hence, we have derived the derivative of sinx using the quotient rule derivative calculator.

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

The easiest way to calculate the derivative sin x 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 this tool.

1. Write the function as sin x 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 x.

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

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

After completing these steps, you will receive the differential of sin x 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 minus sin?

The derivative of minus sine is the measure of rate of change. The derivative of - sin x is - cos x and denoted by d / dx (- sin x) = - cos x.

What is the integral of sin x?

The integral of sin x is -cos x. mathematically, this is written as ∫ sin x dx = -cos x + C, where, C is the integration constant.

What is the formula of sin x?

The formula of sin x can be written as, sin x = cos (π / 2 - x). It is the ratio of the opposite side to the hypotenuse of a triangle.