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

Derivitive of csc(x)

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 in the form f(x) = g(x) / h(x), then the derivative of f(x) with respect to x is given by:

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

In this case, the function f(x) = csc(x) can be written as f(x) = 1/sin(x), where g(x) = 1 and h(x) = sin(x).

Using the quotient rule, we need to find the derivative of g(x), g'(x), and the derivative of h(x), h'(x):

g'(x) = 0 (since g(x) = 1, and the derivative of a constant is always zero)

h'(x) = cos(x) (since the derivative of sin(x) is cos(x))

Now, substituting these values into the quotient rule formula, we have:

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

Since cot(x) = cos(x) / sin(x), we can write the derivative as:

f'(x) = -cot(x) / sin(x)

Therefore, the derivative of csc(x) with respect to x is -cot(x) / sin(x).

More Answers: