How to Find the Derivative of csc(x) with Respect to x Using the Quotient Rule

d/dx csc(x)

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

To find the derivative of csc(x) with respect to x, we will use the quotient rule. The quotient rule states that if we have a function in the form 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

For csc(x), we can rewrite it as 1/sin(x). Applying the quotient rule, we have:

f(x) = 1 / sin(x)
g(x) = 1
h(x) = sin(x)

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

g'(x) = 0 (since g(x) = 1, the derivative of a constant is always zero)
h'(x) = cos(x) (using the derivative of sin(x), which is cos(x))

Plugging these derivatives back into the quotient rule formula, we have:

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

Simplifying the numerator:

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

Now, since csc(x) is equal to 1 / sin(x), we can substitute this back into the expression to obtain:

d/dx csc(x) = -cos(x) / [sin(x)]^2

So, the derivative of csc(x) with respect to x is -cos(x) / [sin(x)]^2.

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