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# Derivative of csc x

TopA branch of mathematics which is deals with the angles and sides of a triangle and also shows the relationship between the angles and sides of triangle is known as Trigonometry. There are so many types of Functions that we study in trigonometry. We calculate several types of Derivatives in this, let’s have a look on derivative of Csc X. As we know, Derivative of Csc X is – csc x cot x. Let’s see it’s prove.
First we write the csc x in the derivative form:

$\frac{d}{dx}$ csc x = - cscx cot x;

We know that csc x = 1 / sin x;

We can also write in place of csc x as:
= $\frac{d}{dx}$ [1 / sin x]
We can solve it by u / v methods:

$(\frac{u}{v})^{1}$ $\frac{vu^{1} - uv^{1}}{v^{2}})$
Put the value of u and v in this expression so that we can easily find the solution.
Here the value of ‘u’ is sin x and the value of ‘v’ is 1, so put the value in the formula:
So we can write the above expression as:

$\frac{d}{dx}$ $\frac{1}{sin x}$ = $\frac{sin x \frac{d}{dx} (1) - 1 \frac{d}{dx} (sin x)}{(sin x)^{2}}$ (1) – 1;
If we find the derivative of 1 and sin x we get:
When we differentiate the number ‘1’ then we get 0, and if we differentiate sin x then we get cos x, so put the derivative value.
= [sin x (0) – 1 (cos x)] / tan2 x;
On further solving this value we get:
= [0 – cos x] / sin2 x;
On further solving we get:
= - csc x cot x.
Thus, we get:
= csc x is – csc x cot x.
This is how we can proof the derivative of Csc X.