Question: 4 Find \( f^{\prime}(x) \) for the function \( f(x)=\frac{\tan x+\csc ^{-1} x-3 \log _{5} x}{\tan ^{-1} x+\sec x-2^{x}} \)

To differentiate the function \( f(x)=\frac{\tan x+\csc^{-1} x-3 \log_{5} x}{\tan^{-1} x+\sec x-2^{x}} \), you can use the quotient rule. Remember to compute the derivative of the numerator and denominator separately. The derivative of \( \tan x \) is \( \sec^2 x \), the derivative of \( \csc^{-1} x \) is \( -\frac{1}{|x|\sqrt{x^2-1}} \), and for logarithms, \( \frac{d}{dx} \log_{b} x = \frac{1}{x \ln b} \). Don’t forget to apply the chain rule when necessary and keep an eye on the rules for differentiating the terms in the denominator as well. Another tip: it's easy to make a mistake with negative signs or forgetting to apply the chain rule, especially under complex functions. Break down the process, simplify each part before putting it all together to avoid confusion. Backtrack if you hit a snag, and remember to double-check your integration and differentiation rules!