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Derivative of Cosec x
The derivative of cosec x is negative of the product of trigonometric functions cosec x and cot x, that is, cosec x cot x. The differentiation of csc x is the process of evaluating the derivative of cosec x with respect to angle x. Before proving the differentiation of cosec x, let us recall the definition of cosec x (also written as csc x). Cosec x is the ratio of the hypotenuse and the perpendicular sides of a rightangled triangle.
Let us understand the differentiation of cosec x along with its proof in different methods such as the first principle of derivatives, chain rule, quotient rule, and also we will solve a few examples using the derivative of cosec x.
1.  What is Derivative of Cosec x? 
2.  Derivative of Cosec x Proof By First Principle 
3.  Derivative of Csc x By Chain Rule 
4.  Derivative of Cosec x By Quotient Rule 
5.  FAQs on Derivative of Cosec x 
What is Derivative of Cosec x?
The differentiation of cosec x with respect to angle x is written as d(cosec x)/dx = (cosec x)' = cot x cosec x. Derivative of cosec x can be calculated using the derivative of sin x. The differentiation of cosec x can be done in different ways. The derivative of cosec x can be derived using the definition of the limit, chain rule, and quotient rule. We use the existing trigonometric identities and existing rules of differentiation to prove the derivative of cosec x to be cot x cosec x.
Derivative of Csc x Formula
The formula for the derivative of cosec x is written as
 d(cosec x)/dx = cot x cosec x
 (cosec x)' = cot x cosec x
Derivative of Cosec x Proof By First Principle
Now, we will derive the derivative of cosec x by the first principle of derivatives, that is, the definition of limits. A derivative is simply a measure of the rate of change. To find the derivative of cosec x, we take the limiting value as x approaches x + h. To simplify this, we set x = x + h, and we want to take the limiting value as h approaches 0. We are going to use certain trigonometry formulas to determine the derivative of csc x. The formulas are:
 cosec x = 1/sin x
 \(\lim_{h\rightarrow 0} \dfrac{( \sin (x+h)\sin x )}{h} = \cos x\)
 cot x = cos x/sin x
\(\begin{align}\frac{\mathrm{d} (\csc x)}{\mathrm{d} x} &= \frac{\mathrm{d} (\frac{1}{\sin x})}{\mathrm{d} x} \\&=\lim_{h\rightarrow 0} \dfrac{\frac{1}{\sin (x+h)}\frac{1}{\sin x}}{(x+h)x} \\&= \lim_{h\rightarrow 0} \dfrac{\frac{\sin x  \sin (x+h)}{\sin x \sin (x+h)}}{h}\\&=\lim_{h\rightarrow 0} \dfrac{\sin x  \sin (x+h) }{h\sin x \sin (x+h)}\\&=\lim_{h\rightarrow 0} \dfrac{( \sin (x+h)\sin x )}{h\sin x \sin (x+h)}\\&=\lim_{h\rightarrow 0} \dfrac{( \sin (x+h)\sin x )}{h} \times \dfrac{1}{\lim_{h\rightarrow 0}\sin x \sin (x+h)}\\&=\cos x \times \dfrac{1}{\sin^2x}\\&=\dfrac{\cos x}{\sin x}\times \dfrac{1}{\sin x}\\&=\cot x \csc x\end{align}\)
Hence, we have derived the derivative of cosec x to be cot x cosec x using the first principle of differentiation.
Derivative of Csc x Proof By Chain Rule
The chain rule for differentiation is: (f(g(x)))’ = f’(g(x)) . g’(x). For this let us note that we can write y = cosec x as y = 1 / (sin x) = (sin x)^{1}. Now, to evaluate the derivative of csc x using the chain rule, we will use certain trigonometric properties and identities such as:
 d(sin x)/dx = cos x
 cos x/ sin x = cot x
We can proceed by using the chain rule.
Given that y = cosec x = 1 / sinx = (sinx)^{1 }[ using trigonometric ratios cosec x = 1/ sin x]
⇒ dy / dx = −(sin x) ^{2 }× [ d (sin x)/ dx ] [ from chain rule ]
⇒ dy / dx = (cos x) × (sin^{2 }x) [ since d (sin x)/ dx = cos x ]
⇒ dy / dx = − (cos x / sin x) × (1 / sin x) [ on distributing the denominator ]
⇒ dy / dx = − (cot x cosec x) [ since cos x / sin x = cot x and 1 / sin x = cosec x ]
Hence, we have derived the derivative of cosec x to be cot x cosec x using the chain rule.
Derivative of Cosec x By Quotient Rule
The quotient rule for differentiation is: (f/g)’ = (f’g  fg’)/g^{2}. To derive the derivative of cosec x, we will use the following formulas:
 d(sin x)/dx = cos x
 cos x /sin x = cot x
 1/sin x = cosec x
To begin with, d(cosec x)/dx = d(1/sin x)/dx
= (1' sin x  (sin x)' 1)/sin^{2}x
= (0. sin x  cos x)/sin^{2}x
= cos x/sin^{2}x
= (cos x/sin x)(1/sin x)
= cot x cosec x
Hence, we have derived the derivative of csc x using the quotient rule.
Important Notes on Derivative of Cosec x
 The derivative of cosec x can be obtained using different methods including the first principle, chain rule and quotient rule.
 Differentiation of cosec x is cot x cosec x.
Topics Related to Derivative of Cosec x
Examples Using Derivative of Cosec x

Example 1: Find the derivative of cosec x cot x.
Solution: The derivative of cosec x cot x can be determined using the product rule. We know that d(cosec x)/dx = cot x cosec x, d(cot x)/dx = cosec^{2}x
Using product rule, we have
d(cosec x cot x)/dx = (cosec x cot x)'
(cosec x)' cot x + cosec x (cot x)'
= cosec x cot x cot x + cosec x (cosec^{2}x)
= cosec x (cot^{2}x + cosec^{2}x)
Answer: Derivative of cosec x cot x is csc x (cot^{2}x + csc^{2}x)

Example 2: Determine the second derivative of cosec x.
Solution: We know that the first derivative of cosec x is cosec x cot x.
To determine the second derivative of cosec x, we differentiate cosec x cot x using the product rule.
Using product rule, we have
(cosec x)'' = (cosec x cot x)'
(cosec x)' cot x + (cosec x) (cot x)'
= cosec x cot x cot x + (cosec x) (cosec^{2}x)
= cosec x (cot^{2}x + cosec^{2}x)
Answer: Hence the second derivative of cosec x is cosec x (cot^{2}x + cosec^{2}x).
FAQs on Derivative of Cosec x
What is the Derivative of Cosec x in Trigonometry?
The derivative of cosec x is negative of the product of trigonometric functions cosec x and cot x, that is, cosec x cot x.
How to Find the Derivative of Cosec x?
Derivative of Cosec x can be calculated using different methods including the first principle of differentiation, quotient rule, and chain rule.
What are the Methods to Prove the Differentiation of Cosec x?
Methods to prove the differentiation of cosec x to be cosec x cot x are:
 Proof by the first principle
 Proof by chain rule
 Proof by quotient rule
What is the Antiderivative of cosec x?
The antiderivative is nothing but the integral of a function. Hence the antiderivative of cosec x is lncosec x + cot x + C.
How to Find the Double Derivative of Csc x?
The double derivative of csc x is csc x (cot^{2}x + csc^{2}x). The double derivative is nothing but the second derivative of a function which can be obtained by differentiating the first derivative of csc x.
Is the Derivative of Cosec x the Same as the Derivative of Cosec Inverse x?
No, the derivative of cosec x is not the same as the derivative of cosec inverse x. The derivative of cosec x is cosec x cot x and the derivative of cosec inverse x is 1/x√(x^{2}1).
What is the Derivative of Csc x in Terms of Cos x and Sin x?
The derivative of csc x is csc x cot x = cos x/sin^{2}x because cot x = cos x/ sin x and csc x = 1/sin x.
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