Derivative of csc(x): Using the Chain Rule and Trigonometric Identities

derivative of cscx

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

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

Recall that the derivative of sin(x) is cos(x). So, we can express csc(x) as 1/sin(x).

Using the chain rule, the derivative of csc(x) can be found as follows:

Let u = sin(x)
Let y = csc(x) = 1/u

dy/dx = dy/du * du/dx

To find dy/du, we differentiate y = 1/u with respect to u.

dy/du = -1/u^2

The derivative of sin(x), du/dx, is cos(x).

Now, we can substitute these values back into the chain rule formula:

dy/dx = dy/du * du/dx
= -1/u^2 * cos(x)

Replacing u with sin(x), we have:

dy/dx = -1/sin^2(x) * cos(x)

Simplifying further, we can use the identity cos^2(x) + sin^2(x) = 1:

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

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

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