If $x Y^2+\frac{x^2}{y}=5$, Then $\frac{d Y}{d X}=$

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Introduction

Implicit differentiation is a powerful technique used in calculus to find the derivative of an implicitly defined function. In this article, we will explore how to use implicit differentiation to solve the given equation $x y2+\frac{x2}{y}=5$ and find the value of $\frac{d y}{d x}$.

Understanding Implicit Differentiation

Implicit differentiation is a method used to find the derivative of an implicitly defined function. An implicitly defined function is a function where the variable is not isolated on one side of the equation. In other words, the variable is not explicitly defined as a function of another variable.

The main idea behind implicit differentiation is to differentiate both sides of the equation with respect to the independent variable, while treating the dependent variable as a function of the independent variable.

Applying Implicit Differentiation to the Given Equation

To apply implicit differentiation to the given equation, we need to differentiate both sides of the equation with respect to x.

ddx(xy2+x2y)=ddx(5)\frac{d}{dx} \left(x y^2+\frac{x^2}{y}\right) = \frac{d}{dx} (5)

Using the product rule and the quotient rule, we can differentiate the left-hand side of the equation as follows:

ddx(xy2)+ddx(x2y)=0\frac{d}{dx} \left(x y^2\right) + \frac{d}{dx} \left(\frac{x^2}{y}\right) = 0

y2+xddx(y2)+2xy−x2y2ddx(y)=0y^2 + x \frac{d}{dx} \left(y^2\right) + \frac{2x}{y} - \frac{x^2}{y^2} \frac{d}{dx} \left(y\right) = 0

Simplifying the Equation

To simplify the equation, we need to use the chain rule to differentiate the terms involving y.

y2+2xyddx(y)+2xy−x2y2ddx(y)=0y^2 + 2xy \frac{d}{dx} \left(y\right) + \frac{2x}{y} - \frac{x^2}{y^2} \frac{d}{dx} \left(y\right) = 0

Combining like terms, we get:

y2+(2xy−x2y2)ddx(y)+2xy=0y^2 + \left(2xy - \frac{x^2}{y^2}\right) \frac{d}{dx} \left(y\right) + \frac{2x}{y} = 0

Solving for $\frac{d y}{d x}$

To solve for $\frac{d y}{d x}$, we need to isolate the term involving $\frac{d y}{d x}$.

(2xy−x2y2)ddx(y)=−y2−2xy\left(2xy - \frac{x^2}{y^2}\right) \frac{d}{dx} \left(y\right) = -y^2 - \frac{2x}{y}

Dividing both sides by $\left(2xy - \frac{x2}{y2}\right)$, we get:

ddx(y)=−y2−2xy2xy−x2y2\frac{d}{dx} \left(y\right) = \frac{-y^2 - \frac{2x}{y}}{2xy - \frac{x^2}{y^2}}

Simplifying the expression, we get:

dydx=−y3−2x2xy2−x2\frac{d y}{d x} = \frac{-y^3 - 2x}{2xy^2 - x^2}

Conclusion

In this article, we used implicit differentiation to solve the given equation $x y2+\frac{x2}{y}=5$ and find the value of $\frac{d y}{d x}$. We applied the product rule and the quotient rule to differentiate the left-hand side of the equation, and then used the chain rule to differentiate the terms involving y. Finally, we isolated the term involving $\frac{d y}{d x}$ and simplified the expression to get the final answer.

Implicit Differentiation: A Powerful Tool for Solving Implicit Equations

Implicit differentiation is a powerful tool for solving implicit equations. It allows us to find the derivative of an implicitly defined function, which can be used to solve a wide range of problems in calculus and other areas of mathematics.

Common Applications of Implicit Differentiation

Implicit differentiation has a wide range of applications in calculus and other areas of mathematics. Some common applications include:

  • Finding the derivative of an implicitly defined function: Implicit differentiation can be used to find the derivative of an implicitly defined function, which can be used to solve a wide range of problems in calculus.
  • Solving implicit equations: Implicit differentiation can be used to solve implicit equations, which can be used to model a wide range of real-world phenomena.
  • Finding the equation of a tangent line: Implicit differentiation can be used to find the equation of a tangent line to a curve, which can be used to model a wide range of real-world phenomena.

Conclusion

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. An implicitly defined function is a function where the variable is not isolated on one side of the equation.

Q: How does implicit differentiation work?

A: Implicit differentiation works by differentiating both sides of the equation with respect to the independent variable, while treating the dependent variable as a function of the independent variable. This involves using the product rule and the quotient rule to differentiate the terms involving the dependent variable.

Q: What are some common applications of implicit differentiation?

A: Some common applications of implicit differentiation include:

  • Finding the derivative of an implicitly defined function
  • Solving implicit equations
  • Finding the equation of a tangent line to a curve

Q: How do I know when to use implicit differentiation?

A: You should use implicit differentiation when you are given an implicitly defined function and you need to find its derivative. This is often the case when you are working with equations that involve multiple variables and you need to find the rate of change of one variable with respect to another.

Q: What are some common mistakes to avoid when using implicit differentiation?

A: Some common mistakes to avoid when using implicit differentiation include:

  • Failing to recognize that the function is implicitly defined
  • Failing to use the product rule and the quotient rule correctly
  • Failing to isolate the term involving the derivative

Q: How do I apply implicit differentiation to a specific problem?

A: To apply implicit differentiation to a specific problem, follow these steps:

  1. Identify the implicitly defined function
  2. Differentiate both sides of the equation with respect to the independent variable
  3. Use the product rule and the quotient rule to differentiate the terms involving the dependent variable
  4. Isolate the term involving the derivative
  5. Simplify the expression to get the final answer

Q: What are some real-world applications of implicit differentiation?

A: Some real-world applications of implicit differentiation include:

  • Modeling the motion of an object under the influence of gravity
  • Modeling the growth of a population
  • Modeling the behavior of a physical system

Q: How do I check my work when using implicit differentiation?

A: To check your work when using implicit differentiation, follow these steps:

  1. Verify that the function is implicitly defined
  2. Check that you have used the product rule and the quotient rule correctly
  3. Check that you have isolated the term involving the derivative
  4. Simplify the expression to get the final answer
  5. Check that the final answer makes sense in the context of the problem

Q: What are some common pitfalls to avoid when using implicit differentiation?

A: Some common pitfalls to avoid when using implicit differentiation include:

  • Failing to recognize that the function is implicitly defined
  • Failing to use the product rule and the quotient rule correctly
  • Failing to isolate the term involving the derivative
  • Failing to simplify the expression to get the final answer

Conclusion

Implicit differentiation is a powerful tool for solving implicit equations. By following the steps outlined in this article, you can apply implicit differentiation to a wide range of problems in calculus and other areas of mathematics. Remember to check your work carefully and avoid common pitfalls to ensure that you get the correct answer.