Derivative of Equations: Understanding and Calculating x^y 2^{x - y}

Understanding and calculating derivatives of complex equations is a fundamental skill in calculus. In this article, we will explore the relationship between xy {2}^{x - y} and how to find its derivative with respect to x.

Introduction to the Equation: x^y 2^{x - y}

Consider the equation xy {2}^{x - y}. This equation is not straightforward and requires a careful approach to solving it. One effective method is logarithmic differentiation, which involves taking the logarithm of both sides of the equation.

Applying Logarithms to Both Sides

Let's start by taking the logarithm of both sides of the equation:

ln({x^y} mo space) ln mo space(2^{x - y})

Using the properties of logarithms, we can rewrite this as:

ln({x^y} mo space) y cdot ln x mo space x - y cdot ln 2

Differentiation with Respect to x

Now, we need to differentiate the equation with respect to x. We use implicit differentiation, which means that we will treat y as a function of x and find dy/dx.

Starting with the left side of the equation:

lnxy y cdot ln x

Differentiating both sides with respect to x:

ln{ x } mo space cdot dfrac{y}{x} cdot dfrac{dy}{dx} y cdot dfrac{1}{x} 1 - dfrac{dy}{dx} cdot ln 2

Simplifying the equation by combining the terms involving dy/dx:

dfrac{dy}{dx} cdot (ln 2 cdot ln x) ln 2 - dfrac{y}{x}

Solving for dy/dx:

dfrac{dy}{dx} dfrac{ln 2 - dfrac{y}{x}}{ln 2 cdot ln x}

Final Formulation

Thus, the derivative of the equation xy {2}^{x - y} with respect to x can be expressed as:

dfrac{dy}{dx} dfrac{x cdot ln 2 - y}{x cdot ln 2 cdot ln x}

Conclusion and Applications

This type of problem is a common example in calculus, particularly when dealing with implicit functions and more complex equations. Understanding and solving these problems helps in developing a strong foundation in calculus, making it easier to tackle more challenging topics in advanced mathematics.

Keywords: derivative, logarithmic differentiation, implicit differentiation