Calculating dy/dx for y ln(x^2) y^2 at Specific Points

When dealing with complex relationships between variables, such as y ln(x^2) y^2, it is crucial to use the techniques of implicit differentiation to find the derivative dy/dx. In this article, we will explore the steps to determine dy/dx at a specific point, such as (10, y_10), and the underlying mathematical principles involved.

Understanding the Problem

The given equation is:

( y ln(x^2) y^2 )

Our goal is to find dy/dx at any point (x, y), and specifically for the point (10, y_10).

Implicit Differentiation

Let's differentiate the equation with respect to x using implicit differentiation. The equation is:

( y ln(x^2) y^2 )

We can apply the chain rule and the rule for differentiating logarithmic functions, which states:

( frac{d}{dx} [ln(u)] frac{1}{u} cdot frac{du}{dx} ) where (u) is a function of (x).

Step-by-Step Differentiation

1. **Differentiate both sides with respect to x:**

( frac{dy}{dx}  frac{1}{x^2} cdot 2x   2y frac{dy}{dx} )

2. **Rearrange the equation to isolate ( frac{dy}{dx} ):

( frac{dy}{dx} - 2y frac{dy}{dx}  frac{2x}{x^2} )

3. **Factor out ( frac{dy}{dx} ):

( frac{dy}{dx} (1 - 2y/x^2)  frac{2x}{x^2} )

4. **Solve for ( frac{dy}{dx} ):

( frac{dy}{dx}  frac{2x}{x^2 - 2yx^2} )

5. **Simplify the expression:

( frac{dy}{dx}  frac{2x}{x^2(1 - 2y/x^2)} )

At the specific point (10, y_10):

( frac{dy}{dx}  frac{2(10)}{10^2(1 - 2y_10/10^2)} )

Given that at x 10, y 2 (as derived from the initial steps), substitute y_10 2:

( frac{dy}{dx}  frac{2(10)}{10^2(1 - 2(2)/10^2)} )

Further simplification:

( frac{dy}{dx}  frac{20}{100(1 - 4/100)} )

( frac{dy}{dx} frac{20}{100 cdot 96/100} )

( frac{dy}{dx} frac{20}{96} )

( frac{dy}{dx} frac{5}{24} )

Conclusion

In conclusion, the derivative ( frac{dy}{dx} ) at the point (10, y_10) for the given equation ( y ln(x^2) y^2 ) is ( frac{5}{24} ).

To further explore similar problems in differentiation or to deepen your understanding of mathematical concepts, continuous practice and application of implicit differentiation with various functions can be highly beneficial. Understanding these fundamental principles is crucial for success in more advanced fields such as engineering, physics, and economics.