What Is The Derivative Of Csc

Jordan Emanuel

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09-12-2024

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The derivative of csc x (cosecant x) is -csc x cot x.

Let's break down how we arrive at this result:

Understanding the Cosecant Function

The cosecant function, csc x, is the reciprocal of the sine function:

csc x = 1 / sin x

Applying the Quotient Rule

To find the derivative of csc x, we use the quotient rule, which states:

d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]²

In our case:

• f(x) = 1 (therefore, f'(x) = 0)
• g(x) = sin x (therefore, g'(x) = cos x)

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

d/dx (csc x) = d/dx (1/sin x) = [(sin x)(0) - (1)(cos x)] / (sin x)²

This simplifies to:

-cos x / (sin x)²

Simplifying the Result

Recall that:

• cos x / sin x = cot x
• 1 / sin x = csc x

Therefore, we can rewrite the derivative as:

-csc x cot x

Conclusion

Therefore, the derivative of csc x is -csc x cot x. Remembering the relationship between cosecant, sine, cosine, and cotangent is key to understanding and deriving this result.