Question

1. find y

(a) y = \sec x + \frac{\tan x}{x}

Explanation:

Step1: Differentiate \(\sec x\)

The derivative of \(\sec x\) is \(\sec x\tan x\), so \(\frac{d}{dx}(\sec x)=\sec x\tan x\).

Step2: Differentiate \(\frac{\tan x}{x}\) using quotient rule

The quotient rule states that if \(y = \frac{u}{v}\), then \(y'=\frac{u'v - uv'}{v^{2}}\). Let \(u = \tan x\) and \(v = x\).

• First, find \(u'\): The derivative of \(\tan x\) is \(\sec^{2}x\), so \(u'=\sec^{2}x\).
• Second, find \(v'\): The derivative of \(x\) is \(1\), so \(v' = 1\).
• Now apply the quotient rule: \(\frac{d}{dx}(\frac{\tan x}{x})=\frac{\sec^{2}x\cdot x-\tan x\cdot1}{x^{2}}=\frac{x\sec^{2}x - \tan x}{x^{2}}\).

Step3: Combine the derivatives

The derivative of \(y=\sec x+\frac{\tan x}{x}\) is the sum of the derivatives of each term:
\(y'=\sec x\tan x+\frac{x\sec^{2}x - \tan x}{x^{2}}\).

Answer:

\(y'=\sec x\tan x+\frac{x\sec^{2}x - \tan x}{x^{2}}\)