Derivative of csc(x) | Step-by-Step Guide and Formula

(d/dx) csc(x)

To find the derivative of csc(x), we can start by expressing csc(x) in terms of sine

To find the derivative of csc(x), we can start by expressing csc(x) in terms of sine. The function csc(x) is defined as the reciprocal of sine:

csc(x) = 1/sin(x)

Now, we can differentiate both sides of this equation with respect to x using the quotient rule. The quotient rule states that the derivative of the quotient of two functions u(x) and v(x) is given by:

(u(x)/v(x))’ = (u'(x)v(x) – u(x)v'(x))/[v(x)]^2

Applying the quotient rule to csc(x) = 1/sin(x), we have:

(csc(x))’ = (1)’sin(x) – 1(cos(x))/(sin(x))^2

The derivative of the constant term 1 is 0, and the derivative of sin(x) is cos(x). Additionally, we can simplify the expression by using the trigonometric identity (cos(x))/(sin(x)) = cot(x). Substituting these values into the equation, we get:

(csc(x))’ = -cot(x)/(sin(x))^2

Therefore, the derivative of csc(x) with respect to x is -cot(x)/(sin(x))^2.

In summary, (d/dx) csc(x) = -cot(x)/(sin(x))^2.

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