91Ó°ÊÓ

Question: Use the Quotient Rule to verify the derivative formula for \(\csc x\), which is \(\frac{d}{d x}(\csc x) = -\csc x \cot x\). Answer: Follow the steps below to verify the derivative formula for \(\csc x\). Step 1: Write csc(x) as a fraction \(\csc x = \frac{1}{\sin x}\) Step 2: Apply the Quotient Rule \(\frac{d}{d x}(\frac{1}{\sin x}) = \frac{-\cos x}{\sin^2 x}\) Step 3: Simplify the expression \(\frac{d}{d x}(\frac{1}{\sin x}) = \frac{-\cos x}{\sin^2 x}\) Step 4: Use trigonometric identities to rewrite the expression \(\frac{-\cos x}{\sin^2 x} = -\csc x \cot x\) Step 5: Conclusion \(\frac{d}{d x}(\csc x)=-\csc x \cot x\) The derivative formula for \(\csc x\) is verified using the Quotient Rule.