If 3 Y 3 − 3 X 2 = − 2 X Y + 3 3y^3 - 3x^2 = -2xy + 3 3 Y 3 − 3 X 2 = − 2 X Y + 3 , Then Find D Y D X \frac{dy}{dx} D X D Y ​ In Terms Of X X X And Y Y Y .

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Introduction

In this article, we will explore the process of finding the derivative of a given function with respect to one of its variables. The given function is a cubic equation involving both $x$ and $y$, and we are asked to find the derivative of $y$ with respect to $x$. This is a classic problem in calculus, and it requires the application of various techniques such as implicit differentiation and the chain rule.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable. In this case, the given function is $3y^3 - 3x^2 = -2xy + 3$. To find the derivative of $y$ with respect to $x$, we need to differentiate both sides of the equation with respect to $x$.

Step 1: Differentiate both sides of the equation

To differentiate both sides of the equation, we need to apply the chain rule and the product rule. The chain rule states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$. The product rule states that if we have a product of two functions of the form $f(x) \cdot g(x)$, then the derivative of this function is given by $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

Applying the chain rule and the product rule to the given equation, we get:

$$\frac{d}{dx}(3y^3) - \frac{d}{dx}(3x^2) = \frac{d}{dx}(-2xy) + \frac{d}{dx}(3)$$

Step 3: Evaluate the derivatives

Evaluating the derivatives on both sides of the equation, we get:

$$9y^2 \cdot \frac{dy}{dx} - 6x = -2x \cdot \frac{dy}{dx} - 2y + 0$$

Step 4: Simplify the equation

Simplifying the equation, we get:

$$9y^2 \cdot \frac{dy}{dx} - 6x = -2x \cdot \frac{dy}{dx} - 2y$$

Step 5: Solve for $\frac{dy}{dx}$

Solving for $\frac{dy}{dx}$, we get:

$$9y^2 \cdot \frac{dy}{dx} + 2x \cdot \frac{dy}{dx} = -2y + 6x$$

$$\frac{dy}{dx} (9y^2 + 2x) = -2y + 6x$$

$$\frac{dy}{dx} = \frac{-2y + 6x}{9y^2 + 2x}$$

Conclusion

In this article, we have found the derivative of the given function with respect to $x$. The derivative is given by $\frac{dy}{dx} = \frac{-2y + 6x}{9y^2 + 2x}$. This is a classic example of implicit differentiation, and it requires the application of various techniques such as the chain rule and the product rule.

Final Answer

The final answer is $\boxed{\frac{-2y + 6x}{9y^2 + 2x}}$.

Related Topics

• Implicit differentiation
• Chain rule
• Product rule
• Derivatives of trigonometric functions
• Derivatives of exponential functions

References

• [1] Calculus by Michael Spivak
• [2] Calculus by James Stewart
• [3] Implicit Differentiation by Paul's Online Math Notes

Keywords

• Implicit differentiation
• Chain rule
• Product rule
• Derivatives
• Calculus
• Mathematics
  

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Q&A

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable. It involves differentiating both sides of an equation with respect to one of the variables, while treating the other variable as a constant.

Q: What is the chain rule?

A: The chain rule is a technique used to find the derivative of a composite function. It states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$.

Q: What is the product rule?

A: The product rule is a technique used to find the derivative of a product of two functions. It states that if we have a product of two functions of the form $f(x) \cdot g(x)$, then the derivative of this function is given by $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

Q: How do we find the derivative of a function using implicit differentiation?

A: To find the derivative of a function using implicit differentiation, we need to differentiate both sides of the equation with respect to one of the variables, while treating the other variable as a constant. We then solve for the derivative of the variable with respect to which we are differentiating.

Q: What is the final answer to the problem?

A: The final answer to the problem is $\frac{dy}{dx} = \frac{-2y + 6x}{9y^2 + 2x}$.

Q: Can we use implicit differentiation to find the derivative of any function?

A: No, we cannot use implicit differentiation to find the derivative of any function. Implicit differentiation is used to find the derivative of a function that is not explicitly defined in terms of one variable. If the function is explicitly defined in terms of one variable, we can use the standard rules of differentiation to find the derivative.

Q: What are some common applications of implicit differentiation?

A: Some common applications of implicit differentiation include:

• Finding the derivative of a function that is defined implicitly
• Finding the derivative of a function that involves trigonometric functions
• Finding the derivative of a function that involves exponential functions
• Finding the derivative of a function that involves logarithmic functions

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

A: Yes, we can use implicit differentiation to find the derivative of a function that involves multiple variables. However, we need to be careful to differentiate both sides of the equation with respect to the correct variable.

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 differentiate both sides of the equation with respect to the correct variable
• Failing to treat the other variable as a constant
• Failing to solve for the derivative of the variable with respect to which we are differentiating
• Failing to check the validity of the solution

Related Topics

• Implicit differentiation
• Chain rule
• Product rule
• Derivatives of trigonometric functions
• Derivatives of exponential functions
• Derivatives of logarithmic functions

References

• [1] Calculus by Michael Spivak
• [2] Calculus by James Stewart
• [3] Implicit Differentiation by Paul's Online Math Notes

Keywords

• Implicit differentiation
• Chain rule
• Product rule
• Derivatives
• Calculus
• Mathematics