Mastering the Quotient Rule: Derivative of csc(x) Simplified

(d/dx) csc(x)

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

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

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

In this case, g(x) = 1 and h(x) = sin(x). Taking the derivatives, we get:

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

h'(x) = cos(x) (since the derivative of sin(x) is cos(x))

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

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

Simplifying, we have:

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

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

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