Find the derivatives of the functions in Exercises 1–42.


𝔂 = 1 x² csc 2
2 x

Verified step by step guidance

1. First, identify the function you need to differentiate: \( y = \frac{1}{x^2} \csc(22x) \). This is a product of two functions: \( \frac{1}{x^2} \) and \( \csc(22x) \).
2. Apply the product rule for differentiation, which states that if you have a function \( u(x) \cdot v(x) \), its derivative is \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \). Here, let \( u(x) = \frac{1}{x^2} \) and \( v(x) = \csc(22x) \).
3. Differentiate \( u(x) = \frac{1}{x^2} \). Use the power rule: \( \frac{d}{dx} x^n = n x^{n-1} \). The derivative of \( \frac{1}{x^2} \) is \( -2x^{-3} \).
4. Differentiate \( v(x) = \csc(22x) \). Use the chain rule and the derivative of \( \csc(x) \), which is \( -\csc(x) \cot(x) \). The derivative of \( \csc(22x) \) is \( -22 \csc(22x) \cot(22x) \) due to the chain rule.
5. Combine the derivatives using the product rule: \( y' = (-2x^{-3}) \cdot \csc(22x) + \frac{1}{x^2} \cdot (-22 \csc(22x) \cot(22x)) \). Simplify the expression to find the derivative.