Differentiation of cosecx

Here you will learn what is the differentiation of cosecx and its proof by using first principle.

Let’s begin –

Differentiation of cosecx

The differentiation of cosecx with respect to x is -cosecx.cotx

i.e. \(d\over dx\) (cosecx) = -cosecx.cotx

Proof Using First Principle :

Let f(x) = cosec x. Then, f(x + h) = cosec(x + h)

\(\therefore\)   \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(f(x + h) – f(x)\over h\)

\(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(cosec(x + h) – cosec x\over h\)

\(\implies\)  \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \({1\over sin(x + h)} – {1\over sin x}\over h\)

\(\implies\)  \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(sin x – sin(x + h)\over h sin x sin(x +h)\)

By using trigonometry formula,

[sin C – sin D = \(2sin ({C – D\over 2})cos ({C + D\over 2})\)]

\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(2sin ({x – x – h\over 2})cos({x + x + h\over 2})\over h sin x sin (x + h)\)

\(\implies\) \(d\over dx\)(f(x)) = \(lim_{h\to 0}\) \(2sin ({-h\over 2})cos({x + h/2})\over h sin x sin (x + h)\)

\(\implies\) \(d\over dx\)(f(x)) = -\(lim_{h\to 0}\) \(cos ({x + h/2})\over sin x sin(x + h)\).\(lim_{h\to 0}\) \(sin(h/2)\over (h/2)\)

because, [\(lim_{h\to 0}\)\(sin(h/2)\over (h/2)\) = 1]

\(\implies\) \(d\over dx\)(f(x)) = -\(cos x\over sin x sin x\)(1) = -cot x cosec x

Hence, \(d\over dx\) (cosec x) = =cosecx.cotx

Example : What is the differentiation of cosec x + x with respect to x?

Solution : Let y = cosec x + x

\(d\over dx\)(y) = \(d\over dx\)(cosec x + x)

\(\implies\) \(d\over dx\)(y) = \(d\over dx\)(cosec x) + \(d\over dx\)(x)

By using cosecx differentiation we get,

\(\implies\) \(d\over dx\)(y) = -cosec x cot x + 1

Hence, \(d\over dx\)(sec x + x) = -cosec x cot x + 1

Example : What is the differentiation of \(cosec\sqrt{x}\) with respect to x?

Solution : Let y = \(cosec\sqrt{x}\)

\(d\over dx\)(y) = \(d\over dx\)(\(cosec\sqrt{x}\))

By using chain rule we get,

\(\implies\) \(d\over dx\)(y) = \(1\over 2\sqrt{x}\)(\(-cosec \sqrt{x}.cot\sqrt{x}\))

Hence, \(d\over dx\)(\(cosec\sqrt{x}\)) = -\(1\over 2\sqrt{x}\)(\(cosec \sqrt{x}.cot\sqrt{x}\))


Related Questions

What is the Differentiation of cosec inverse x ?

What is the Integration of Cosecx ?

What is the differentiation of 1/sinx ?

Leave a Comment

Your email address will not be published.