A Step-by-Step Guide on Finding the Derivative of the Cosecant Function using the Quotient Rule

derivative of cosecant

To find the derivative of the cosecant function, we can use the quotient rule

To find the derivative of the cosecant function, we can use the quotient rule. The derivative of cosecant (csc) of x, denoted as csc(x), can be determined as follows:

Let us assume that y = csc(x), where x is the independent variable and y is the dependent variable.

Step 1: Express csc(x) in terms of sine (sin(x)):
csc(x) = 1/sin(x)

Step 2: Apply the quotient rule, which states that if y = u/v, where u and v are functions of x, then the derivative dy/dx is given by:
dy/dx = (v * du/dx – u * dv/dx) / v^2

In our case, let u = 1 and v = sin(x).

Step 3: Calculate the derivatives of u and v:
du/dx = 0 (since the derivative of a constant is always zero)
dv/dx = cos(x) (derivative of sin(x))

Step 4: Substituting the values into the quotient rule formula:
dy/dx = (sin(x)*0 – 1*cos(x)) / sin^2(x)
dy/dx = -cos(x) / sin^2(x)

Step 5: Simplify the expression:
dy/dx = -cos(x) / (1 – cos^2(x))
dy/dx = -1 / (sin(x) * (1/sin^2(x)))
dy/dx = -1 / (sin(x) / sin^2(x))
dy/dx = -1 / (1 / sin(x))
dy/dx = -sin(x)

Thus, the derivative of cosecant (csc(x)) is equal to -sin(x).

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