How to Find the Derivative of csc(x) using the Quotient Rule

d/dx 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 the derivative of f(x)/g(x) is equal to (f'(x)g(x) – f(x)g'(x))/(g(x))^2.

In this case, f(x) is equal to 1 and g(x) is equal to sin(x), since csc(x) is equal to 1/sin(x).

Using the quotient rule, we can calculate the derivative of csc(x) as follows:

First, let’s find f'(x) and g'(x):

f'(x) = derivative of 1 = 0
g'(x) = derivative of sin(x) = cos(x)

Now, substitute these values into the quotient rule:

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

Simplifying further, we can use the identity 1 + cot^2(x) = csc^2(x) to write -cos(x)/sin^2(x) as -1/sin(x) * cos(x)/sin(x) = -cot(x) * csc(x).

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

More Answers: