Derivative of csc(x) with respect to x: An in-depth guide using the quotient rule and trigonometric identities

d/dx [csc x]

To find the derivative of csc(x) with respect to x (denoted as d/dx [csc x]), we can use the quotient rule for differentiation

To find the derivative of csc(x) with respect to x (denoted as d/dx [csc x]), we can use the quotient rule for differentiation.

The derivative of csc(x) is given by:

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

Where cot(x) is the cotangent function.

Let’s break down the steps for the derivative:

Step 1: Recall that csc(x) is equal to 1/sin(x).

Step 2: Apply the quotient rule, which states that the derivative of a quotient of two functions is given by (g * f’ – f * g’) / g^2. In this case, f(x) = 1 and g(x) = sin(x).

So, using the quotient rule for csc(x), we have:

d/dx [csc x] = (1 * sin'(x) – sin(x) * 1′) / sin^2(x)

Step 3: Calculate the derivative of sin(x). The derivative of sin(x) is cos(x). Hence, sin'(x) = cos(x).

Now, substituting these values, we get:

d/dx [csc x] = (sin'(x) – sin(x) * 1′) / sin^2(x)
= (cos(x) – sin(x) * 0) / sin^2(x)
= cos(x) / sin^2(x)

Step 4: We can now simplify the expression cos(x) / sin^2(x) further. Recall that for trigonometric identities involving sine and cosine, sin^2(x) + cos^2(x) = 1. So, we can substitute 1 – cos^2(x) for sin^2(x).

d/dx [csc x] = cos(x) / sin^2(x)
= cos(x) / (1 – cos^2(x))

This is the simplified form of the derivative of csc(x) with respect to x, i.e., d/dx [csc x] = cos(x) / (1 – cos^2(x)).

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