Derivative of \( \displaystyle \frac{\sqrt[3]{x}}{\sec{\left(x \right)}} \)

Problem 2.935 · hard

1. \[ \frac{d}{d x} \frac{\sqrt[3]{x}}{\sec{\left(x \right)}} \]
   derivative rewriteStart with the derivative of the function. Rewrite the reciprocal of sec(x) as cos(x).✓ Proved
2. \[ = \frac{d}{d x} \sqrt[3]{x} \cos{\left(x \right)} \]
   algebraSimplify the expression.✓ Proved
3. \[ = \sqrt[3]{x} \frac{d}{d x} \cos{\left(x \right)} + \cos{\left(x \right)} \frac{d}{d x} \sqrt[3]{x} \]
   productApply the product rule.✓ Proved
4. \[ = \sqrt[3]{x} \frac{d}{d x} \cos{\left(x \right)} + \frac{\cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]
   powerDifferentiate the first term using the power rule.✓ Proved
5. \[ = - \sqrt[3]{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]
   trig algebraDifferentiate the second term. Simplify the expression.✓ Proved
6. \[ = \frac{- 3 x \sin{\left(x \right)} + \cos{\left(x \right)}}{3 x^{\frac{2}{3}}} \]
   simplifyCombine into a single fraction.✓ Proved