91Ó°ÊÓ

Trigonometric functions play a crucial role in calculus, serving as fundamental building blocks for many mathematical expressions. The cosecant function, \( \csc x \), is one such example. It's defined as the reciprocal of the sine function, \( \csc x = \frac{1}{\sin x} \). Differentiating trigonometric functions often involves memorizing their derivative formulas. For \( \csc x \), its derivative is \( -\csc x \cot x \). This derivative comes from the reciprocal rule and the chain rule in differentiation.
Understanding trigonometric identities and their derivatives is key when handling problems involving these functions. Remember, consistent practice with these types of problems will enhance your intuition for them.

• \( \csc x = \frac{1}{\sin x} \)
• Derivative of \( \csc x = -\csc x \cot x \)

The power rule is a fundamental differentiation rule in calculus, which states that the derivative of \( x^n \) is \( nx^{n-1} \). This rule is widely used because it simplifies the process of finding derivatives of power functions.
In our exercise, the term \( -4 \sqrt{x} \) is rewritten as \( -4x^{1/2} \) to apply the power rule. When differentiating, multiplying the exponent by the coefficient (-4 times 1/2), gives us \( -2 \). Adjusting the exponent by subtracting one results in \( x^{-1/2} \). Thus, the derivative becomes \( -2x^{-1/2} \).

• Original term: \( x^{n} \)
• Power rule: \( \frac{d}{dx}(x^n) = nx^{n-1} \)
• Example: \( \frac{d}{dx}(-4x^{1/2}) = -2x^{-1/2} \)

Differentiation rules are principles that allow us to find the derivative, or rate of change, of a function. Several key rules simplify this process:

• The **constant rule** states that the derivative of a constant is 0. For example, \( y = 7 \) results in \( \frac{d}{dx}(7) = 0 \).
• The **sum or difference rule** lets us differentiate each term individually in a function, then add or subtract these derivatives as needed. Consider the function \( y = \csc x - 4\sqrt{x} + 7 \):
  • Differentiate each term separately: \(-\csc x \cot x\) for \( \csc x \), \(-2x^{-1/2}\) for \(-4\sqrt{x}\), and \(0\) for the constant \(7\).
  • Combine the results: \( \frac{dy}{dx} = -\csc x \cot x - \frac{2}{\sqrt{x}} \).

These foundational rules streamline the process of finding derivatives, making complex problems more manageable.