Differentiate. \[ y=\frac{4 x}{5-\tan (x)} \] \( y^{\prime}=\square \) Need Help? Read it Watch It \( \square \) Master It

To differentiate the function \( y = \frac{4x}{5 - \tan(x)} \), you'll want to use the quotient rule, which states that if you have a function of the form \( \frac{u}{v} \), then its derivative is given by \( \frac{u'v - uv'}{v^2} \). Here, \( u = 4x \) and \( v = 5 - \tan(x) \). Now apply the derivative: - \( u' = 4 \) - \( v' = -\sec^2(x) \) (since the derivative of \( -\tan(x) \) is \( -\sec^2(x) \)). Now, plugging in these results into the quotient rule formula, you have: \[ y' = \frac{(4)(5 - \tan(x)) - (4x)(-\sec^2(x))}{(5 - \tan(x))^2} \] With some simplification, your final answer will look something like: \[ y' = \frac{20 - 4\tan(x) + 4x\sec^2(x)}{(5 - \tan(x))^2} \] This expression captures the rate of change of your original function. Enjoy your math journey! One fascinating tidbit: The tangent function is deeply intertwined with not just trigonometry but also calculus as it exhibits periodic behavior, which can lead to interesting properties in integrals and series. And when tackling derivatives, be sure to double-check the rules: the quotient rule can be tricky! A common mistake is to forget to differentiate both the numerator and denominator appropriately. So always take it step by step and write out what each part represents!