If \(y = (\tan x)^x\), then find \(\frac{dy}{dx}\).
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Solution and Explanation
The given function is: \[ y = (\tan x)^x. \]
Take the natural logarithm on both sides to simplify the power: \[ \ln y = x \ln (\tan x). \]
Differentiate both sides with respect to \(x\): \[ \frac{1}{y} \cdot \frac{dy}{dx} = \ln (\tan x) + x \cdot \frac{1}{\tan x} \cdot \sec^2 x. \]
Simplify: \[ \frac{1}{y} \cdot \frac{dy}{dx} = \ln (\tan x) + x \cdot \frac{\sec^2 x}{\tan x}. \]
Multiply through by \(y = (\tan x)^x\): \[ \frac{dy}{dx} = (\tan x)^x \left[\ln (\tan x) + x \cdot \frac{\sec^2 x}{\tan x}\right]. \]
Simplify further: \[ \frac{dy}{dx} = (\tan x)^x \left[\ln (\tan x) + x \cdot \frac{1}{\sin x \cos x}\right]. \]
Final answer: \[ \frac{dy}{dx} = (\tan x)^x \left[\ln (\tan x) + x \cdot \csc x \sec x\right]. \]
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R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
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