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. The chain rule states that if we have a composition of functions, then the derivative of that composition is the derivative of the outer function multiplied by the derivative of the inner function.
In this case, we have the function f(x) = csc(x), which can also be written as f(x) = 1/sin(x).
Let’s find the derivative by using the chain rule:
1. Rewrite f(x) = 1/sin(x).
2. Let u = sin(x), then f(x) = 1/u.
3. Now, we can find the derivative of f(x) with respect to u: f'(u) = -1/u^2.
4. Next, find the derivative of u with respect to x: du/dx = cos(x).
5. Finally, using the chain rule, we have:
f'(x) = f'(u) * du/dx
= -1/u^2 * cos(x).
Substituting back u = sin(x), we get:
f'(x) = -1/(sin(x))^2 * cos(x).
Simplifying further, we can rewrite this as:
f'(x) = -cos(x)/sin^2(x).
Therefore, the derivative of csc(x) is -cos(x)/sin^2(x).
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