The Derivative of csc(x): A Step-by-Step Guide to Finding the Derivative of the Cosecant Function

Derivative of csc(x)

To find the derivative of csc(x), we can start by expressing csc(x) in terms of sine:

csc(x) = 1/sin(x)

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

To find the derivative of csc(x), we can start by expressing csc(x) in terms of sine:

csc(x) = 1/sin(x)

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

g'(x) = 0 (since g(x) = 1, the derivative is 0)

h'(x) = cos(x) (this is the derivative of sin(x), which is cos(x))

Now, we can 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

Since f(x) = csc(x), we can rewrite the derivative as:

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

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

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