Calculating the Derivative of csc(x) Using the Quotient Rule in Math

d/dx (csc x)

To find the derivative of the function f(x) = csc(x), where csc(x) represents the cosecant of x, we can apply the quotient rule

To find the derivative of the function f(x) = csc(x), where csc(x) represents the cosecant of x, we can apply the quotient rule. Recall that the quotient rule states:
If f(x) = g(x) / h(x), then f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2.

In the case of f(x) = csc(x), we can rewrite it as f(x) = 1 / sin(x), since csc(x) is the reciprocal of sin(x). Now, let’s apply the quotient rule:

g(x) = 1
h(x) = sin(x)

Finding the derivatives:

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

Now we can substitute these values into the quotient rule formula:

f'(x) = (0 * sin(x) – 1 * cos(x)) / (sin(x))^2
= -cos(x) / sin^2(x)
= -cos(x) / sin(x) * sin(x)
= -cos(x) / sin(x) * 1 / sin(x)
= -cot(x) / sin(x)

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

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