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 right-angled 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.

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

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.

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.

The quotient rule for differentiation is: (f/g)’ = (f’g - fg’)/g². 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)/sin²x

= (0. sin x - cos x)/sin²x

= -cos x/sin²x

= (-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

• Derivative of cos x
• Derivative of sin inverse x
• Derivative of cot x