Finding the Derivative of csc(x): Step-by-Step Guide and Formula

Dx {csc (x)}=?

To find the derivative of csc(x), we can start by expressing it as a reciprocal of sine function:

csc(x) = 1/sin(x)

Now, let’s find the derivative using the quotient rule:

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

f'(x) = 0 and g'(x) = cos(x)

Using the quotient rule, the derivative of csc(x) is given by:

(csc(x))’ = (f'(x)g(x) – f(x)g'(x))/(g(x))^2

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

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

Alternatively, we can rewrite csc(x) as:

csc(x) = (1/sin(x)) * (sin(x)/sin(x))

= (sin(x))^(-1) * (sin(x)/sin(x))

= (sin(x))^(-1)

Taking the derivative using the chain rule, we have:

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

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

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

To find the derivative of csc(x), we can start by expressing it as a reciprocal of sine function:

csc(x) = 1/sin(x)

Now, let’s find the derivative using the quotient rule:

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

f'(x) = 0 and g'(x) = cos(x)

Using the quotient rule, the derivative of csc(x) is given by:

(csc(x))’ = (f'(x)g(x) – f(x)g'(x))/(g(x))^2

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

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

Alternatively, we can rewrite csc(x) as:

csc(x) = (1/sin(x)) * (sin(x)/sin(x))

= (sin(x))^(-1) * (sin(x)/sin(x))

= (sin(x))^(-1)

Taking the derivative using the chain rule, we have:

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

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

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

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