How to Find the Derivative of the Trigonometric Function csc(x) with Respect to x

d/dx csc(x)

To find the derivative of the trigonometric function csc(x) with respect to x, we can start by expressing csc(x) in terms of sine

To find the derivative of the trigonometric function csc(x) with respect to x, we can start by expressing csc(x) in terms of sine. The reciprocal of sine is csc(x), so we can rewrite csc(x) as 1/sin(x).

Now, using the quotient rule of differentiation, we can find the derivative of csc(x) as follows:

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

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

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

Now, let’s find each component of this derivative.

d/dx (1) = 0, because the derivative of a constant is always zero.

d/dx (sin(x)) is the derivative of sine, which is cosine (cos(x)).

Substituting these values into the quotient rule formula, we have:

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

This can also be written as -cot(x) / sin(x). Therefore, the derivative of csc(x) with respect to x is -cot(x) / sin(x).

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