How to Find the Derivative of csc(x) Using the Chain Rule and Quotient Rule

d/dx csc(x)

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

To find the derivative of csc(x), we can use the chain rule. The chain rule states that if we have a function of a function, the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.

Let’s consider csc(x) as the outer function and x as the inner function. The derivative of the outer function, csc(x), can be found using the quotient rule.

The quotient rule states that if we have a function of the form f(x)/g(x), the derivative is given by (g(x)f'(x) – f(x)g'(x))/[g(x)]^2.

In the case of csc(x), we can rewrite it as 1/sin(x). Now we can apply the quotient rule.

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

f'(x) = 0 (the derivative of a constant is always 0)
g'(x) = cos(x) (the derivative of sin(x) is cos(x))

Now we can plug these values into the quotient rule formula:

(csc(x))’ = [(sin(x)(0) – 1(cos(x)))] / [sin(x)]^2
= -cos(x) / sin(x)^2

Simplifying this expression, we can rewrite it using the identity cos(x)/sin(x) = cot(x):

(csc(x))’ = -cot(x) / sin(x)
= -cot(x) csc(x)

Therefore, the derivative of csc(x) is -cot(x) csc(x).

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