Using the Chain Rule to Find the Derivative of csc(x) Mathematically

d/dx(csc(x))

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

To find the derivative of the function f(x) = csc(x), where csc(x) represents the cosecant of x, we can use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), the derivative of f(g(x)) with respect to x is given by the derivative of f with respect to g, multiplied by the derivative of g with respect to x.

Let’s break down the function f(x) = csc(x) into its component parts:

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

Now, we can differentiate the function using the chain rule.

First, let’s find the derivative of 1/sin(x). To do this, we can rewrite 1/sin(x) as sin(x)^(-1).

Using the power rule, the derivative of sin(x)^(-1) with respect to x is:

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

Now, we need to find the derivative of sin(x). The derivative of sin(x) is cos(x).

Substituting this back into the previous equation, we have:

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

Finally, we simplify this expression:

d/dx(csc(x)) = (-cos(x))/sin^2(x)

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

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