Introduction to the Derivative of sin(2x)
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of sin2x can be calculated by following the rules of differentiation. Or, we can directly find the differentiation of sin2x by applying the first principle of differentiation.
In this article, you will learn what the derivative of sin (2x) is and how to differentiate sin2x by using different approaches.
What is the derivative of sin2x?
The differential of the trigonometric function sin(2x) with respect to the variable cx is denoted by d/dx(sin(2x)), which evaluates to 2cos(2x). This represents the rate of change of sin(2x). In a triangle, the differential of sin2x corresponds to the ratio of the side opposite to the angle x and the hypotenuse. Mathematically, it is expressed as;
sin x = opposite side / hypotenuse
Derivative of sin(2x) formula
The formula for finding the derivative of sin(2x) is 2cos(2x), which means that the rate of change of sin(2x) with respect to x is equal to 2cos(2x). This can be represented mathematically as;
d/dx(sin(2x)) = 2cos(2x).
How do you prove the sin2x derivative?
There are various methods to derive derivatives of sin (2x). Therefore, we can prove the sin2x differentiation by using;
First Principle
Chain Rule
Quotient Rule
Each method provids a different way to compute the derivative sin2x. By using these methods, we can mathematically prove the formula for finding differentiation of sin2x.
Sin 2x derivative by first principle
The first principle method, also known as the delta method, refers to finding the general expression for the slope of a curve by using algebra. This method is often used to calculate the derivative of a function. The derivative is a measure of the instantaneous rate of change of a function, which can be defined mathematically as the limit of the difference quotient as h approaches 0, where h is a small positive number. This can be represented as;
f(x)=lim f(x+h)-f(x)/h
Proof of derivative of sin(2x) by first principle
To prove the derivative of sin (2x) by using the first principle, replace f(x) by sin (2x) or you can replace it by sin x to calculate derivative of sin x.
f′(x)=limh➜0f(x+h)-f(x)/h
f(x) = lim sin 2(x+h) - sin (2x)/h
Therefore,
f(x) = lim [sin 2(x+h) - sin (2x)]/h
Now, by the trigonometric formula, sin A cos B + cos A sin B = sin (A + B)
f(x) = lim [sin 2x cos 2h + cos 2x sin 2h - sin 2x]/h
f(x) = lim [- sin 2x(1 - cos 2h) + cos 2x sin 2h]/h
Now, by using the half-angle formula, 1- cos 2h = 2 sin2 (h), the above equation is written as:
f(x) = (-sin 2x) { lim [(2 sin2 (h))]/h} + (cos 2x) {lim (sin 2h)/2h}
f(x) =(-sin 2x) [lim (sin(h))/(h). lim sin (h)] + (cos 2x) {lim (sin 2h)/2h}
As we know,
Lim (sin 2x/2x) = 2, we get
f(x) = 0+cos2x (2)
Hence
f(x) = 2cos (2x)
Derivative of sin2x by chain rule
To calculate the sin2x 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 formula of chain rule calculator is defined as;
dy/dx = dy/du x du/dx
Proof of sin2x differentiation by chain rule
To prove the sin 2x derivative by using chain rule, we start by assuming that sin (2x) can be written as the combination of two functions. Using this let us find the derivative of sin(2x)
y = sin u where u = 2x
Using chain rule,
y = cos u.du/dx
and
du/dx = 2
Now, using the value of u.
y = 2cos (2x)
Thus, we have derived the formula of derivative of sin (2x) by chain rule.
Derivative of sin(2x) using quotient rule
Another method for finding the derivative of sin 2x 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 formula of derivative quotient rule calculator is defined as:
d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2
Proof of derivative of sin(2x) by quotient rule
To prove the derivative of sin (2x), we can start by writing it,
f(x) = sin (2x) = 1/ cosec (2x) =u/v
Supposing that u = 1 and v = cosec (2x). Now by quotient rule,
f(x) = (vu - uv)/v2
f'(x) = [cosec (2x) d/dx(1) + 1. d/dx(cosec (2x))] / (cosec 2x)2
= [cosec x (0) - 1(-2cosec 2x cot 2x)] / cosec2 2x
= (2cosec (2x).cot (2x)) / cosec2 2x
= 2cos (2x)
Hence, we have derived the derivative of sin2x using the quotient rule of differentiation.
How to find the differentiation of sin2x with a calculator?
The easiest way to calculate the derivative of sin 2x 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.
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 sin2x differentiation.
Now, select the variable by which you want to differentiate sin2x. Here you have to choose x.
Select how many times you want to differentiate sin 2x. In this step, you can choose 2 for second and 3 to find the third derivative calculator.
Click on the calculate button.
After completing these steps, you will receive the derivative of sin 2x 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 sin 3x?
The derivative of sin 3x is equal to the 3 cos 3x that is the same as the derivative of sin (2x).
How differentiation of sinx is cosx?
The formula for the derivative of sinx cosx is given by, d(sinx cosx)/dx or (sinx cosx)' = cos2x. The derivative of a function is the slope of the tangent to the function at the point of contact.
What is the formula of sin (2x)?
The formula of sin x can be written as, sin 2x = 2sin x.cos x. It is the product of 2 sin x and cos x.