Understanding the Derivative of csc²(x) and its Application in Calculus

csc²(x) is the derivative of?

To determine the derivative of csc²(x), we need to apply the rules of differentiation

To determine the derivative of csc²(x), we need to apply the rules of differentiation.

First, let’s rewrite csc²(x) in terms of the sine function:
csc²(x) = (1 / sin(x))²

Next, to find the derivative, we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))²

Now, let g(x) = 1 and h(x) = sin(x). Taking the derivatives, we have:
g'(x) = 0 (the derivative of a constant is zero)
h'(x) = cos(x) (the derivative of sin(x) is cos(x))

Substituting these values into the quotient rule formula, we have:
csc²(x)’ = ((0 * sin(x)) – (1 * cos(x))) / (sin(x))²
= (-cos(x)) / (sin²(x))

Simplifying further, we can write csc²(x)’ as:
csc²(x)’ = -cos(x) / sin²(x)
= -cot(x) / sin(x)

Therefore, the derivative of csc²(x) is -cot(x) / sin(x).

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