Question

1. $y = 3csc x+\frac{5}{x}$

Explanation:

Step1: Differentiate \(3\csc x\)

The derivative of \(\csc x\) is \(-\csc x\cot x\). Using the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\), the derivative of \(3\csc x\) is \(3\times(-\csc x\cot x)=- 3\csc x\cot x\).

Step2: Differentiate \(\frac{5}{x}\)

Rewrite \(\frac{5}{x}\) as \(5x^{-1}\). Using the power rule \((x^n)^\prime=nx^{n - 1}\), when \(n=-1\), the derivative of \(5x^{-1}\) is \(5\times(-1)x^{-1 - 1}=-5x^{-2}=-\frac{5}{x^{2}}\).

Step3: Sum the derivatives

By the sum rule \((f(x)+g(x))^\prime=f^\prime(x)+g^\prime(x)\), the derivative of \(y = 3\csc x+\frac{5}{x}\) is \(y^\prime=-3\csc x\cot x-\frac{5}{x^{2}}\).

Answer:

\(y^\prime=-3\csc x\cot x-\frac{5}{x^{2}}\)