Some standard derivatives: Derivative of csc function

CSC is the acronym for cosecant function. Csc is the reciprocal of sine function. Symbolically it can be written as : csc (x) = 1/sin(x).

Derivative of csc function:
The derivative of csc function is the slope of tangent to the curve of the equation y = csc (x) at any point x. It can also be called the gradient of the csc function. To find the derivative of the function csc (x) we use the quotient rule which is as follows:
If a function f is such that it is a quotient of two functions g and h, symbolically,
f(x) = h(x)/g(x), then the derivative of f, represented by f’(x) is given by the formula,
f’(x) = [g(x)*h’(x) – h(x)*g’(x)]/(g(x))^2
Using the above rule we can find the derivative of csc function as follows:
(d/dx) csc (x) = (d/dx) (1/sin x) = [sin x * (d/dx)(1) – 1* (d/dx) (sin x)]/sin ^2 (x)
= -cos x/sin^2(x)
= -csc x cot x
Therefore derivative of csc is –csc x cot x.

Derivative of csc^-1 (x):
Inverse of the cosecant function is written as csc^(-1) (x). The derivative of the inverse of csc function can be found as follows:
Let y = csc^-1 (x), |x| >1
Thus, x = csc y, y belongs to (0,pi) – {pi/2}
So, dx/dy = - csc y cot y ? 0 because csc y ? 0 and since y belongs to (0,pi) – {pi/2}, cot y ? 0
Therefore, dy/dx = -1/(csc y cot y)
Since y belongs to (0,pi) – {pi/2} that means, either y belongs to (0,pi/2) or y belongs to (pi/2 , pi)
In both cases the following holds.
Then x = csc y > 0 and so |x| = x
Also cot y > 0 and so cot y = sqrt(csc^2 (x) – 1) = sqrt(x^2 - 1)
Therefore dy/dx = -1/x*sqrt(x^2 – 1)
Thus derivative of csc^-1 (x) = -1/x*sqrt(x^2 – 1)

Derivative of csc^squared (x):
Derivative of csc^2 (x) can be found using the chain rule.
Let y = csc^2 (x) and let csc x = u
Then y = u^2
Therefore dy/du = 2u
And since u  = csc x
So, du/dx = -csc x  cot x
Thus dy/dx = dy/du * du/dx
= 2u * (-csc x cot x)
= 2*csc x * (-csc x cot x)
= -2 csc^2 (x) cot x
Thus we see that derivative of csc^2 (x) = -2 csc^2 (x) cot x