An In-Depth Explanation of How to Find the Derivative of csc x

Derivative of csc x

To find the derivative of csc x, we can use the quotient rule with the expression:

csc x = 1/sin x

Let’s proceed with the differentiation:

First, we need to find the derivatives of sin x and 1:

d(sin x)/dx = cos x

To find the derivative of csc x, we can use the quotient rule with the expression:

csc x = 1/sin x

Let’s proceed with the differentiation:

First, we need to find the derivatives of sin x and 1:

d(sin x)/dx = cos x … (1) (derivative of sin x)

d(1)/dx = 0 … (2) (derivative of a constant)

Now, applying the quotient rule, if we have f(x) = g(x)/h(x), then the derivative is given by:

f'(x) = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2

In the case of csc x = 1/sin x, we have:

f'(x) = [(1 * sin x) – (1 * cos x)] / [sin x]^2
= (sin x – cos x) / [sin x]^2

Therefore, the derivative of csc x is:

d(csc x)/dx = (sin x – cos x) / [sin x]^2

This is the final answer.

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