The Derivative of Cosecant Function (csc(x)): Formula and Explanation

d/dx csc(x)

To find the derivative of csc(x), also known as the cosecant function, we can use the quotient rule

To find the derivative of csc(x), also known as the cosecant function, we can use the quotient rule.

The formula for the derivative of a function y = f(x) = csc(x) is given by:

dy/dx = -csc(x)cot(x)

To see how this is derived, let’s start with the basic trigonometric identity:

csc(x) = 1/sin(x)

Now, let’s find the derivative using the quotient rule. The quotient rule states that for a function of the form y = f(x) / g(x), the derivative is given by:

dy/dx = (f'(x)g(x) – g'(x)f(x)) / [g(x)]^2

In this case, f(x) = 1 and g(x) = sin(x). Taking the derivatives of these functions, we have:

f'(x) = 0 (since it is a constant)
g'(x) = cos(x) (the derivative of sin(x) is cos(x))

Plugging these values into the quotient rule formula, we get:

dy/dx = (0 * sin(x) – cos(x) * 1) / [sin(x)]^2
= -cos(x) / [sin(x)]^2

Since cos(x) / [sin(x)]^2 is equal to cot(x) (the cotangent function), we have:

dy/dx = -csc(x)cot(x)

Therefore, the derivative of csc(x) with respect to x is -csc(x)cot(x).

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