Find D Y D X \frac{dy}{dx} D X D Y ​ By Implicit Differentiation For The Equation X 2 − 10 X Y + Y 2 = 10 X^2 - 10xy + Y^2 = 10 X 2 − 10 X Y + Y 2 = 10 .

Introduction

Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. It is a powerful tool that allows us to differentiate functions that are not easily differentiated using other methods. In this article, we will use implicit differentiation to find the derivative of the function $y$ with respect to $x$ for the equation $x^2 - 10xy + y^2 = 10$.

The Equation

The given equation is $x^2 - 10xy + y^2 = 10$. This is an implicit equation, meaning that it is not easily solved for $y$ in terms of $x$. However, we can still use implicit differentiation to find the derivative of $y$ with respect to $x$.

Implicit Differentiation

To find the derivative of $y$ with respect to $x$ using implicit differentiation, we will differentiate both sides of the equation with respect to $x$. This will give us:

$$\frac{d}{dx}(x^2 - 10xy + y^2) = \frac{d}{dx}(10)$$

Using the chain rule and the product rule, we can differentiate each term on the left-hand side:

$$\frac{d}{dx}(x^2) - \frac{d}{dx}(10xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(10)$$

This gives us:

$$2x - 10\left(x\frac{dy}{dx} + y\right) + 2y\frac{dy}{dx} = 0$$

Solving for $\frac{dy}{dx}$

Now, we need to solve for $\frac{dy}{dx}$. To do this, we will isolate the term involving $\frac{dy}{dx}$ on one side of the equation. First, we will move the $2x$ term to the right-hand side:

$$-10\left(x\frac{dy}{dx} + y\right) + 2y\frac{dy}{dx} = -2x$$

Next, we will factor out the $\frac{dy}{dx}$ term:

$$-10x\frac{dy}{dx} - 10y + 2y\frac{dy}{dx} = -2x$$

Now, we will combine like terms:

$$-10x\frac{dy}{dx} + 2y\frac{dy}{dx} = -2x + 10y$$

Next, we will factor out the $\frac{dy}{dx}$ term:

$$\frac{dy}{dx}(-10x + 2y) = -2x + 10y$$

Finally, we will divide both sides by $-10x + 2y$ to solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx} = \frac{-2x + 10y}{-10x + 2y}$$

Simplifying the Expression

We can simplify the expression for $\frac{dy}{dx}$ by factoring out a $-2$ from the numerator and a $-10$ from the denominator:

$$\frac{dy}{dx} = \frac{-2(x - 5y)}{-10(x - y)}$$

Now, we can cancel out the common factor of $-2$:

$$\frac{dy}{dx} = \frac{x - 5y}{5(x - y)}$$

Final Answer

The final answer is $\frac{dy}{dx} = \frac{x - 5y}{5(x - y)}$. This is the derivative of the function $y$ with respect to $x$ for the equation $x^2 - 10xy + y^2 = 10$.

Conclusion

$$$$$$$$

Introduction

Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. In our previous article, we used implicit differentiation to find the derivative of the function $y$ with respect to $x$ for the equation $x^2 - 10xy + y^2 = 10$. In this article, we will answer some common questions related to implicit differentiation.

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. It is a powerful tool that allows us to differentiate functions that are not easily differentiated using other methods.

Q: How do I use implicit differentiation?

A: To use implicit differentiation, you need to differentiate both sides of the equation with respect to $x$. This will give you an equation involving the derivative of $y$ with respect to $x$. You can then solve for the derivative of $y$ with respect to $x$.

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

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

• Not differentiating both sides of the equation with respect to $x$
• Not isolating the term involving the derivative of $y$ with respect to $x$
• Not checking for extraneous solutions

Q: Can I use implicit differentiation to find the derivative of a function that is not implicitly defined?

A: No, implicit differentiation is only used to find the derivative of an implicitly defined function. If the function is not implicitly defined, you can use other methods such as the power rule, product rule, and quotient rule to find the derivative.

Q: How do I check for extraneous solutions when using implicit differentiation?

A: To check for extraneous solutions, you need to plug the solution back into the original equation and check if it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: Can I use implicit differentiation to find the derivative of a function that has multiple variables?

A: Yes, implicit differentiation can be used to find the derivative of a function that has multiple variables. However, you need to be careful when differentiating the function and isolating the term involving the derivative of one of the variables.

Q: How do I use implicit differentiation to find the derivative of a function that has a trigonometric function?

A: To use implicit differentiation to find the derivative of a function that has a trigonometric function, you need to use the chain rule and the product rule. You also need to be careful when differentiating the trigonometric function and isolating the term involving the derivative of one of the variables.

Q: Can I use implicit differentiation to find the derivative of a function that has a logarithmic function?

A: Yes, implicit differentiation can be used to find the derivative of a function that has a logarithmic function. However, you need to be careful when differentiating the logarithmic function and isolating the term involving the derivative of one of the variables.

Conclusion

Implicit differentiation is a powerful tool in calculus that allows us to differentiate functions that are not easily differentiated using other methods. In this article, we answered some common questions related to implicit differentiation. We hope that this article has been helpful in understanding implicit differentiation and how to use it to find the derivative of an implicitly defined function.

Additional Resources

• Implicit Differentiation Tutorial
• Implicit Differentiation Examples
• Implicit Differentiation Practice Problems

Final Answer

The final answer is that implicit differentiation is a powerful tool in calculus that allows us to differentiate functions that are not easily differentiated using other methods.