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: \[ \frac{dy}{dx} = (\tan x)^x \left[\ln (\tan x) + x \cdot \csc x \sec x\right]. \]
The given graph illustrates:
On the basis of the following hypothetical data, calculate the percentage change in Real Gross Domestic Product (GDP) in the year 2022 – 23, using 2020 – 21 as the base year.
Year | Nominal GDP | Nominal GDP (Adjusted to Base Year Price) |
2020–21 | 3,000 | 5,000 |
2022–23 | 4,000 | 6,000 |