Finding the Derivative of csc(x) Using the Quotient Rule

(d/dx) csc(x)

To find the derivative of csc(x) with respect to x, we can use the quotient rule

To find the derivative of csc(x) with respect to x, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:

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

In this case, csc(x) can be expressed as 1/sin(x). So let’s use the quotient rule to find its derivative.

Let g(x) = 1 and h(x) = sin(x).

Using the quotient rule, we have:

(csc(x))’ = [(1)’ * sin(x) – 1 * (sin(x))’] / [sin(x)]^2

The derivative of 1 is 0, and the derivative of sin(x) is cos(x). Plugging these values into the equation, we get:

(csc(x))’ = [0 * sin(x) – 1 * cos(x)] / [sin(x)]^2

Simplifying further, it becomes:

(csc(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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