Find The Derivative Of The Function \[$ Y \$\] Defined Implicitly In Terms Of \[$ X \$\].$\[ 9xy^3 + 4x^3y = 1 \\]\[$\frac{dy}{dx} = \, \square\$\]
Implicit Differentiation: A Step-by-Step Guide to Finding the Derivative of a Function Defined Implicitly
Implicit differentiation is a powerful technique used to find the derivative of a function defined implicitly in terms of another variable. In this article, we will explore the concept of implicit differentiation and apply it to find the derivative of the function defined implicitly in terms of x.
What is Implicit Differentiation?
Implicit differentiation is a method used to find the derivative of a function that is defined implicitly in terms of another variable. In other words, it is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable. This technique is particularly useful when dealing with functions that are defined implicitly in terms of x and y.
The Concept of Implicit Differentiation
Implicit differentiation is based on the concept of 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)) * g'(x). The product rule states that if we have a function of the form f(x) * g(x), then the derivative of this function is given by f'(x) * g(x) + f(x) * g'(x).
Applying Implicit Differentiation to the Given Function
The given function is defined implicitly in terms of x and y as follows:
9xy^3 + 4x^3y = 1
To find the derivative of this function, we will apply the concept of implicit differentiation. We will differentiate both sides of the equation with respect to x, treating y as a function of x.
Step 1: Differentiate Both Sides of the Equation
To differentiate both sides of the equation, we will apply the product rule and the chain rule. The derivative of the left-hand side of the equation is given by:
d(9xy^3)/dx + d(4x^3y)/dx = d(1)/dx
Using the product rule, we can rewrite the derivative of the left-hand side of the equation as:
9(y^3 + 3xy^2(dy/dx)) + 4(3x^2y + x^3(dy/dx)) = 0
Step 2: Simplify the Equation
To simplify the equation, we will combine like terms and isolate the term containing dy/dx.
9y^3 + 27xy^2(dy/dx) + 12x^2y + 4x^3(dy/dx) = 0
Step 3: Isolate the Term Containing dy/dx
To isolate the term containing dy/dx, we will move all the terms containing dy/dx to one side of the equation.
27xy^2(dy/dx) + 4x^3(dy/dx) = -9y^3 - 12x^2y
Step 4: Factor Out the Term Containing dy/dx
To factor out the term containing dy/dx, we will use the distributive property.
dy/dx(27xy^2 + 4x^3) = -9y^3 - 12x^2y
Step 5: Solve for dy/dx
To solve for dy/dx, we will divide both sides of the equation by the coefficient of dy/dx.
dy/dx = (-9y^3 - 12x^2y) / (27xy^2 + 4x^3)
In this article, we have applied the concept of implicit differentiation to find the derivative of the function defined implicitly in terms of x. We have differentiated both sides of the equation, simplified the equation, isolated the term containing dy/dx, factored out the term containing dy/dx, and solved for dy/dx. The final answer is:
dy/dx = (-9y^3 - 12x^2y) / (27xy^2 + 4x^3)
- Find the derivative of the function defined implicitly in terms of x:
x2y3 + 2xy^2 = 1
- Find the derivative of the function defined implicitly in terms of x:
3xy^2 + 4x^2y = 1
- To find the derivative of the function defined implicitly in terms of x, we will apply the concept of implicit differentiation.
d(x2y3)/dx + d(2xy^2)/dx = d(1)/dx
Using the product rule, we can rewrite the derivative of the left-hand side of the equation as:
2xy^3 + 3x2y2(dy/dx) + 4y^2 + 4xy(dy/dx) = 0
Simplifying the equation, we get:
2xy^3 + 3x2y2(dy/dx) + 4y^2 + 4xy(dy/dx) = 0
Isolating the term containing dy/dx, we get:
3x2y2(dy/dx) + 4xy(dy/dx) = -2xy^3 - 4y^2
Factoring out the term containing dy/dx, we get:
dy/dx(3x2y2 + 4xy) = -2xy^3 - 4y^2
Solving for dy/dx, we get:
dy/dx = (-2xy^3 - 4y^2) / (3x2y2 + 4xy)
- To find the derivative of the function defined implicitly in terms of x, we will apply the concept of implicit differentiation.
d(3xy^2)/dx + d(4x^2y)/dx = d(1)/dx
Using the product rule, we can rewrite the derivative of the left-hand side of the equation as:
6xy + 6x^2y(dy/dx) + 8xy + 8x^2(dy/dx) = 0
Simplifying the equation, we get:
6xy + 6x^2y(dy/dx) + 8xy + 8x^2(dy/dx) = 0
Isolating the term containing dy/dx, we get:
6x^2y(dy/dx) + 8x^2(dy/dx) = -6xy - 8xy
Factoring out the term containing dy/dx, we get:
dy/dx(6x^2y + 8x^2) = -6xy - 8xy
Solving for dy/dx, we get:
dy/dx = (-6xy - 8xy) / (6x^2y + 8x^2)
The final answer is:
dy/dx = (-9y^3 - 12x^2y) / (27xy^2 + 4x^3)
Note: The final answer is the same as the answer obtained in the example problem 1.
Implicit Differentiation: A Q&A Guide
Implicit differentiation is a powerful technique used to find the derivative of a function defined implicitly in terms of another variable. In this article, we will explore the concept of implicit differentiation and answer some frequently asked questions about this topic.
Q: What is implicit differentiation?
A: Implicit differentiation is a method used to find the derivative of a function that is defined implicitly in terms of another variable. In other words, it is a technique used to find the derivative of a function that is not explicitly defined in terms of one variable.
Q: When is implicit differentiation used?
A: Implicit differentiation is used when we have a function that is defined implicitly in terms of x and y, and we need to find the derivative of this function with respect to x.
Q: How do I apply implicit differentiation?
A: To apply implicit differentiation, we need to differentiate both sides of the equation with respect to x, treating y as a function of x. We will then isolate the term containing dy/dx and solve for dy/dx.
Q: What are the steps involved in implicit differentiation?
A: The steps involved in implicit differentiation are:
- Differentiate both sides of the equation with respect to x, treating y as a function of x.
- Simplify the equation and isolate the term containing dy/dx.
- Factor out the term containing dy/dx.
- Solve for dy/dx.
Q: What are some common mistakes to avoid when applying implicit differentiation?
A: Some common mistakes to avoid when applying implicit differentiation include:
- Not differentiating both sides of the equation with respect to x.
- Not treating y as a function of x.
- Not isolating the term containing dy/dx.
- Not factoring out the term containing dy/dx.
Q: Can implicit differentiation be used to find the derivative of a function that is defined explicitly in terms of x?
A: No, implicit differentiation cannot be used to find the derivative of a function that is defined explicitly in terms of x. Implicit differentiation is used to find the derivative of a function that is defined implicitly in terms of x and y.
Q: What are some real-world applications of implicit differentiation?
A: Some real-world applications of implicit differentiation include:
- Finding the rate of change of a quantity that is defined implicitly in terms of another variable.
- Modeling the behavior of a system that is defined implicitly in terms of another variable.
- Solving optimization problems that involve implicit functions.
Q: Can implicit differentiation be used to find the derivative of a function that is defined in terms of multiple variables?
A: Yes, implicit differentiation can be used to find the derivative of a function that is defined in terms of multiple variables. However, the process of implicit differentiation becomes more complex when dealing with multiple variables.
Q: What are some tips for mastering implicit differentiation?
A: Some tips for mastering implicit differentiation include:
- Practicing implicit differentiation with simple examples.
- Using the chain rule and the product rule to simplify the equation.
- Isolating the term containing dy/dx and solving for dy/dx.
- Checking your work by plugging in the value of x and y into the original equation.
Implicit differentiation is a powerful technique used to find the derivative of a function defined implicitly in terms of another variable. By following the steps involved in implicit differentiation and avoiding common mistakes, you can master this technique and apply it to a wide range of problems.