𝑑/𝑑𝑥[csc 𝑥]
To find the derivative of csc(x) with respect to x, we can use the chain rule
To find the derivative of csc(x) with respect to x, we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), then the derivative of f(g(x)) with respect to x is given by:
(dy/dx) = (dy/dg) * (dg/dx)
In this case, our function is f(g) = csc(g), where g(x) = x.
First, we need to find the derivative of g(x) = x with respect to x. The derivative of x with respect to x is simply 1.
So, (dg/dx) = 1.
Next, we need to find the derivative of csc(g) = csc(x) with respect to g. The derivative of csc(g) with respect to g is equal to -csc(g) * cot(g). In this case, g(x) = x, so we substitute x in place of g.
Thus, (dy/dg) = -csc(x) * cot(x).
Finally, we multiply these two derivatives together:
(dy/dx) = (dy/dg) * (dg/dx)
= (-csc(x) * cot(x)) * 1
= -csc(x) * cot(x)
Therefore, the derivative of csc(x) with respect to x is -csc(x) * cot(x).
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