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

ddx cscx

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 of the form f(x) = g(x)/h(x), the derivative is given by:

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

In the case of csc(x), we can rewrite it as 1/sin(x). Therefore, g(x) = 1 and h(x) = sin(x).

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = d/dx (1) = 0 (the derivative of a constant is always 0)

h'(x) = d/dx (sin(x)) = cos(x) (the derivative of sin(x) is cos(x))

Now, let’s substitute these values into the quotient rule formula:

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

Simplifying further:

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

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

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