How to Find the Derivative of csc(x) and Use the Chain Rule in Calculus

d/dx[cscx]

To find the derivative of csc(x) with respect to x, we can start by using the chain rule

To find the derivative of csc(x) with respect to x, we can start by using the chain rule.

The chain rule states that if we have a composite function f(g(x)), the derivative is the product of the derivative of f with respect to g multiplied by the derivative of g with respect to x.

In this case, our function is csc(x). We can rewrite it as f(g(x)) where f(u) = csc(u) and g(x) = x.

Now we need to find the derivatives of f(u) and g(x).

The derivative of f(u) = csc(u) can be found using the quotient rule. The quotient rule states that if we have a function of the form f(u) = g(u)/h(u), the derivative is given by f'(u) = (g'(u)h(u) – g(u)h'(u))/[h(u)]^2.

In this case, g(u) = 1 and h(u) = sin(u). Taking their derivatives, we have g'(u) = 0 and h'(u) = cos(u).

Now we can calculate the derivative of f(u) = csc(u):

f'(u) = (g'(u)h(u) – g(u)h'(u))/[h(u)]^2
= (0*sin(u) – 1*cos(u))/[sin(u)]^2
= -cos(u)/[sin(u)]^2
= -cot(u)csc(u)

Now, we can calculate the derivative of csc(x) using the chain rule:

d/dx [csc(x)] = f'(g(x)) * g'(x)
= -cot(g(x))csc(g(x)) * g'(x)

Since g(x) = x, we have:

d/dx [csc(x)] = -cot(x)csc(x) * 1
= -cot(x)csc(x)

Therefore, the derivative of csc(x) with respect to x is -cot(x)csc(x).

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