How to Find the Derivative of csc(x) with Respect to x Using the Quotient Rule

d/dx(csc(x))

To find the derivative of csc(x) with respect to x, we can use the chain rule and the fact that csc(x) is equal to 1/sin(x)

To find the derivative of csc(x) with respect to x, we can use the chain rule and the fact that csc(x) is equal to 1/sin(x).

Let’s start by writing csc(x) as 1/sin(x):

csc(x) = 1/sin(x)

Now, let’s use the quotient rule to find the derivative:

d/dx(csc(x)) = d/dx(1/sin(x))

Using the quotient rule, the derivative of 1/sin(x) with respect to x is given by:

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

Now, let’s calculate the derivatives:

The derivative of 1 with respect to x is 0, because 1 is a constant.

The derivative of sin(x) with respect to x is cos(x), according to the chain rule.

Substituting these derivative values into the quotient rule formula, we get:

= (sin(x)(0) – 1(cos(x))) / (sin^2(x))

Simplifying further, we have:

= -cos(x) / (sin^2(x))

So, the derivative of csc(x) with respect to x is equal to -cos(x) divided by the square of sin(x).

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