If $ {{x}^{y}}.{{y}^{x}}=16, $ then $ \frac{dy}{dx} $ at (2, 2) is
- 1
- 2
- $ -1 $
- $ -2 $
The Correct Option is C
Solution and Explanation
Taking log on both sides,
$ y\text{ }log\text{ }x+x\text{ }log\text{ }y=log\text{ }16 $
Differentiating on both sides,
$ \frac{y}{x}+\log x\frac{dy}{dx}+\frac{x}{y}\frac{dy}{dx}+\log y=0 $
$ \left( \frac{x}{y}+\log x \right)\frac{dy}{dx}=-\left( \frac{y}{x}+\log y \right) $
$ \frac{dy}{dx}=-\frac{y}{x}\frac{(y+x\log y)}{(x+y\log x)} $
$ {{\left( \frac{dy}{dx} \right)}_{at(2,2)}}=\frac{-2}{2}\left( \frac{2+2\log 2}{2+2\log 2} \right)=-1 $
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Concepts Used:
Differential Equations
A differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on.
Orders of a Differential Equation
First Order Differential Equation
The first-order differential equation has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as: dy/dx = f(x, y) = y’
Second-Order Differential Equation
The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Types of Differential Equations
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
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