## d/dx(cscx)

### 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.

First, let’s rewrite csc(x) as 1/sin(x):

csc(x) = 1/sin(x)

Now, to find the derivative, we need to differentiate 1/sin(x) with respect to x. Using the quotient rule, the derivative will be:

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

The derivative of 1 is zero, so we can simplify it further:

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

Now, let’s handle the derivative of sin(x):

d/dx(sin(x)) = cos(x)

Substituting this into our equation, we get:

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

Therefore, the derivative of csc(x) with respect to x is:

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

Alternatively, using trigonometric identities, we can rewrite csc(x) as:

csc(x) = 1/sin(x) = sin^(-1)(x)

Then, using the power rule for differentiation, the derivative becomes:

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

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