How to Find the Derivative of the Cosecant Function using the Chain Rule

d/dx(cscx)

To find the derivative of the function f(x) = csc(x), we can use the chain rule

To find the derivative of the function f(x) = csc(x), we can use the chain rule.

The chain rule states that if we have a composition of functions, u = g(x) and y = f(u), then the derivative of y with respect to x is given by the product of the derivative of y with respect to u and the derivative of u with respect to x.

In this case, we have f(x) = csc(x), where csc(x) represents the cosecant function. Recall that the cosecant function is defined as the reciprocal of the sine function: csc(x) = 1/sin(x).

To calculate the derivative of f(x) = csc(x), we will convert it to a composition of functions. Let u(x) = sin(x). Therefore, f(x) = (1/u(x)). Applying the chain rule, we can find the derivative as follows:

f'(x) = (1/u(x))’ = (-1/u^2(x)) * u'(x)

To find u'(x), we differentiate sin(x) using the chain rule. Since the derivative of sin(x) is cos(x), u'(x) = cos(x).

Now, substituting the values we obtained into our derivative equation:

f'(x) = (-1/u^2(x)) * u'(x)
= (-1/u^2(x)) * cos(x)

Since u(x) = sin(x), we replace u^2(x) with sin^2(x):

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

Therefore, the derivative of f(x) = csc(x) is f'(x) = (-1/sin^2(x)) * cos(x).

Note: This derivative can also be expressed using trigonometric identities. By using the identity csc(x) = 1/sin(x), we can simplify the derivative as f'(x) = -cot(x)csc(x).

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