Derivative of csc x
To find the derivative of the cosecant function, denoted as csc(x), we can use the chain rule
To find the derivative of the cosecant function, denoted as csc(x), we can use the chain rule. The chain rule states that if we have a composition of functions, then the derivative is given by the derivative of the outer function multiplied by the derivative of the inner function.
The cosecant function can be stated as the reciprocal of the sine function: csc(x) = 1/sin(x).
Let’s begin by finding the derivative of sin(x):
d(sin(x))/dx = cos(x)
Next, we can find the derivative of csc(x) using the chain rule:
d(csc(x))/dx = d(1/sin(x))/dx
To apply the chain rule, we need to find the derivative of the outer function (1/u), where u is the sine function in this case, and multiply it by the derivative of the inner function (sin(x)).
Using the quotient rule, we can find the derivative of the outer function:
d(1/u)/dx = -u’/u^2
Here, u’ represents the derivative of the inner function, which is cos(x) in this case, and u^2 represents the square of the sine function.
Plugging in the values, we have:
d(csc(x))/dx = -cos(x)/(sin(x))^2
So, the derivative of csc(x) is -cos(x)/(sin(x))^2.
It is worth mentioning that the cosecant function is not defined for x values where sin(x) equals zero. At these points, the derivative of csc(x) is undefined.
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