The Derivative of Cosecant Function | Quotient Rule and Chain Rule Explained

derivative of cscx

The derivative of csc(x), where csc(x) represents the cosecant function of x, can be found using the quotient rule and chain rule

The derivative of csc(x), where csc(x) represents the cosecant function of x, can be found using the quotient rule and chain rule.

The cosecant function is defined as csc(x) = 1/sin(x). Therefore, we can rewrite csc(x) as csc(x) = (1/sin(x)).

To find the derivative of csc(x), we use the quotient rule, which states that if we have a function f(x) = g(x)/h(x), then the derivative is given by:

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

Applying the quotient rule to csc(x) = (1/sin(x)), we have:

csc'(x) = ((1)’ * sin(x) – 1 * (sin(x))’)/(sin(x))^2.

Since (1)’ is equal to 0, we have:

csc'(x) = (-sin(x))/(sin(x))^2.

To simplify further, we can rewrite sin(x) as 1/sin(x), since sin(x) = 1/sin(x). Therefore, we have:

csc'(x) = (-1)/(sin(x))^2.

Lastly, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to rewrite (sin(x))^2 as 1 – (cos(x))^2. Hence, the derivative of csc(x) is:

csc'(x) = -(1/(1 – (cos(x))^2)).

Therefore, the derivative of csc(x) is -(1/(1 – (cos(x))^2)) or equivalently, -cot(x)csc(x).

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