The Quotient Rule: Finding the Derivative of csc(x)

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 use the quotient rule

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

The quotient rule gives us the formula for finding the derivative of a function f(x) = g(x)/h(x) as:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / [h(x)]^2

In this case, g(x) = 1 and h(x) = sin(x), since csc(x) is equal to 1/sin(x).

Using this information, let’s find the derivative of f(x) = csc(x):

g'(x) = 0 (the derivative of a constant is zero)

h'(x) = cos(x) (the derivative of sin(x) is cos(x) using the chain rule)

Plugging these values into the quotient rule formula:

f'(x) = (0 * sin(x) – 1 * cos(x)) / [sin(x)]^2

Simplifying:

f'(x) = -cos(x) / (sin(x))^2

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

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