Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.
d/dx (csc x) = -csc x cot x

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Start by recalling the definition of the cosecant function: \( \csc x = \frac{1}{\sin x} \). We need to find the derivative of this function with respect to \( x \).

Apply the Quotient Rule for derivatives, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{v \cdot u' - u \cdot v'}{v^2} \). Here, \( u = 1 \) and \( v = \sin x \).

Calculate the derivatives: \( u' = 0 \) since the derivative of a constant is zero, and \( v' = \cos x \) since the derivative of \( \sin x \) is \( \cos x \).

Substitute these into the Quotient Rule formula: \( \frac{\sin x \cdot 0 - 1 \cdot \cos x}{(\sin x)^2} = \frac{-\cos x}{\sin^2 x} \).

Recognize that \( \frac{-\cos x}{\sin^2 x} \) can be rewritten using trigonometric identities as \( -\csc x \cdot \cot x \), thus verifying the derivative formula \( \frac{d}{dx}(\csc x) = -\csc x \cot x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quotient Rule

The Quotient Rule is a fundamental technique in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be calculated using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential for verifying derivatives of functions like csc(x), which can be expressed as a quotient.

Cosecant Function

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). Understanding the properties and behavior of the cosecant function is crucial for differentiating it, as it influences the application of the Quotient Rule. Additionally, knowing the relationship between csc(x) and sin(x) helps in simplifying the derivative calculations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables involved. Key identities, such as sin^2(x) + cos^2(x) = 1, are often used in calculus to simplify expressions and derivatives. Familiarity with these identities is important when verifying derivative formulas, as they can help in transforming and simplifying the resulting expressions from the differentiation process.

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