Introduction to the Derivative of csc 3x
Derivatives have a wide range of applications in almost every field of engineering and science. The derivative of -sin x can be calculated by following the rules of differentiation. Or, we can directly find the derivative of csc (3x) by applying the first principle of differentiation. In this article, you will learn what the derivative of csc 3x is and how to calculate the derivative of csc3x by using different approaches.
What is the derivative of cosec 3x?
The derivative of csc(3x) with respect to x can be found using the formula d/dx(csc(3x)) = -3csc(3x)cot(3x).
This formula shows that the derivative of csc(3x) is related to the cotangent function, which is the reciprocal of the tangent function. In other words, the derivative of csc(3x) is the negative of the product of csc(3x) and cot(3x).
The tangent function, on the other hand, is the slope of a line at a given point on a curve. In trigonometry, it is defined as the ratio of the sine and cosine functions, which are related to the sides of a right triangle. Specifically, csc x, which is the reciprocal of sin x, represents the ratio of the hypotenuse to the opposite side of the triangle.
Derivative of csc(3x) formula
The formula of derivative of csc3x is equal to the negative of the product of csc(3x) and cot(3x), that is;
d/dx (csc(3x)) = -3csc (3x).cot (3x)
This formula is derived using the chain rule of differentiation and the trigonometric identities for csc(3x) and cot(3x). Also, the trigonometric differentiation is used to derive the derivatives of trigonometric identities like sin, cos, tan etc.
How do you prove the derivative of csc(3x)?
There are multiple ways to derive the csc 3x derivative. The three common methods used for finding the derivative of csc(3x) are;
- First Principle
- Chain Rule
- Quotient Rule
Each method provides a different way to compute the csc 3x derivative. By using these methods, we can mathematically prove the formula for finding derivatives of csc3x.
Derivative of csc(3x) by first principle
According to the first principle of derivative, the csc 3x derivative is equal to -3csc(3x)cot(3x). 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 f(x+h)-f(x) / h
This formula allows us to determine the rate of change of a function at a specific point by using limit definition of derivative.
Proof of derivative of cosec 3x by first principle
To prove the derivative of csc (3x) by using the first principle, replace f(x) by csc (3x).
f'(x)=lim h➜0f(x+h)-f(x)/h
f'(x) = lim tan 3(x+h) - csc (3x)/h
Therefore,
f'(x) = lim [csc 3(x+h) - csc (3x)]/h
Now, by the trigonometric formula, csc x = 1/sin x. So,
f'(x) = lim [1/sin 3(x+h) - 1/sin (3x)]/h
f'(x) = lim [sin (3x) - sin 3(x+h)]/hsin 3(x+h) sin 3x
f'(x) = lim - [sin 3(x+h) - sin 3x]/hsin 3(x+h)sin 3x
Now,
f'(x) = - lim[sin 3(x+h) - sin 3x]/h. lim 1/[sin 3(x+h)sin 3x]
Since lim[sin 3(x+h) - sin 3x]/h = cos 3x
f'(x) = - cos (3x). lim 1/[sin 3(x+h)sin (3x)]
f'(x) = - cos (3x). 1/sin3 (3x)
Hence the derivative of csc (3x) is,
f'(x) = - 3csc (3x).cot (3x)
The derivative of csc(2x) can be verified by using first principles.
Derivative of csc3x by chain rule
To calculate the derivative of csc (3x), 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 csc(3x) by chain rule
To prove the derivative of csc (3x) by using chain rule, assume that csc (3x) can be written as the combination of two functions. Using this let us find the derivative of csc(3x)
y = csc u where u = 3x
Using chain rule,
y = -csc u.cot u.du/dx
and
du/dx = 3
Now, using the value of u.
y = -3csc (3x).cot (3x)
Thus, you can also derive the formula of derivative of cosec 3x by chain rule calculator.
Derivative of csc(3x) using quotient rule
Another method for finding the csc 3x differentiation is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of cosecant can also be calculated using the quotient rule. The quotient rule is defined as:
d/dx (f/g) = f(x). g(x) -g(x).f(x) /{g(x)}2
Proof of derivative of csc(3x) by quotient rule
To prove the derivative of csc (3x), we can start by writing it as,
f(x) = csc (3x) = 1/sin (3x) =u/v
Supposing that u = 1 and v = sin (3x). Now by quotient rule,
f(x) = (vu - uv)/v2
f'(x) = [sin (3x) d/dx(1) - d/dx(sin (3x))] / (cos 3x)2
= [0 - (3cos (3x))] / sin2 3x
= [-3cos (3x)]/ sin2 3x
= -3csc (3x).cot (3x)
Hence, we have derived the derivative of csc3x using the quotient rule of derivative calculator.
How to find the derivative of csc(3x) with a calculator?
The easiest way to calculate the derivative of csc (3x) is by using an online tool. You can use our dy/dx calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.
- Write the function as csc(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 csc(3x).
- Now, select the variable by which you want to differentiate csc3x. Here you have to choose x.
- Select how many times you want to differentiate csc 3x. In this step, you can choose 3 for the second derivative, 3 for the third derivative and so on.
- Click on the calculate button. After this step, you will get the derivative of csc(3x)within a few seconds.
After completing these steps, you will receive the derivative of csc3x 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 Cos square 3x?
The derivative of cos squared 3x can be calculated by using the power rule of differentiation. Therefore the derivative of cos square 3x is -3sin(6x).
Where is Csc X undefined?
Since cosecant is the reciprocal of sine, any angle x for which sin x = 0 must have an undefined cosecant because it would have a denominator of zero. Since sin(0) has a value of 0, the cosecant of 0 must be undefinable.
Is csc the same as cosec?
One of the six trigonometric ratios is known as cosecant and can also be written as cosec or csc. In a right triangle, the cosecant formula is the length of the hypotenuse divided by the length of the other side.